Calculus Examples

$\int \frac{x + 1}{{(x^{2} + 2 x - 3)}^{2}} d x$

Step 1

Write the fraction using partial fraction decomposition.

Tap for more steps...

Step 1.1

Decompose the fraction and multiply through by the common denominator.

Tap for more steps...

Step 1.1.1

Factor the fraction.

Tap for more steps...

Step 1.1.1.1

Factor $x^{2} + 2 x - 3$ using the AC method.

Tap for more steps...

Step 1.1.1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 1.1.1.1.2

Write the factored form using these integers.

$\frac{x + 1}{{((x - 1) (x + 3))}^{2}}$

Step 1.1.1.2

Apply the product rule to $(x - 1) (x + 3)$ .

$\frac{x + 1}{{(x - 1)}^{2} {(x + 3)}^{2}}$

Step 1.1.2

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $A$ .

$\frac{A}{{(x - 1)}^{2}}$

Step 1.1.3

$\frac{A}{{(x - 1)}^{2}} + \frac{B}{x - 1}$

Step 1.1.4

$\frac{A}{{(x - 1)}^{2}} + \frac{B}{x - 1} + \frac{C}{{(x + 3)}^{2}}$

Step 1.1.5

$\frac{A}{{(x - 1)}^{2}} + \frac{B}{x - 1} + \frac{C}{{(x + 3)}^{2}} + \frac{D}{x + 3}$

Step 1.1.6

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is ${(x - 1)}^{2} {(x + 3)}^{2}$ .

$\frac{(x + 1) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x - 1)}^{2} {(x + 3)}^{2}} = \frac{(A) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x - 1)}^{2}} + \frac{(B) {(x - 1)}^{2} {(x + 3)}^{2}}{x - 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.7

Cancel the common factor of ${(x - 1)}^{2}$ .

Tap for more steps...

Step 1.1.7.1

Cancel the common factor.

Step 1.1.7.2

Rewrite the expression.

$\frac{(x + 1) {(x + 3)}^{2}}{{(x + 3)}^{2}} = \frac{(A) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x - 1)}^{2}} + \frac{(B) {(x - 1)}^{2} {(x + 3)}^{2}}{x - 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.8

Cancel the common factor of ${(x + 3)}^{2}$ .

Tap for more steps...

Step 1.1.8.1

Cancel the common factor.

Step 1.1.8.2

Divide $x + 1$ by $1$ .

$x + 1 = \frac{(A) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x - 1)}^{2}} + \frac{(B) {(x - 1)}^{2} {(x + 3)}^{2}}{x - 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9

Simplify each term.

Tap for more steps...

Step 1.1.9.1

Cancel the common factor of ${(x - 1)}^{2}$ .

Tap for more steps...

Step 1.1.9.1.1

Cancel the common factor.

$x + 1 = \frac{A {(x - 1)}^{2} {(x + 3)}^{2}}{{(x - 1)}^{2}} + \frac{(B) {(x - 1)}^{2} {(x + 3)}^{2}}{x - 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.1.2

Divide $(A) {(x + 3)}^{2}$ by $1$ .

$x + 1 = (A) {(x + 3)}^{2} + \frac{(B) {(x - 1)}^{2} {(x + 3)}^{2}}{x - 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.2

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$x + 1 = A ((x + 3) (x + 3)) + \frac{(B) {(x - 1)}^{2} {(x + 3)}^{2}}{x - 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.3

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Tap for more steps...

Step 1.1.9.3.1

Apply the distributive property.

$x + 1 = A (x (x + 3) + 3 (x + 3)) + \frac{(B) {(x - 1)}^{2} {(x + 3)}^{2}}{x - 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.3.2

Apply the distributive property.

$x + 1 = A (x \cdot x + x \cdot 3 + 3 (x + 3)) + \frac{(B) {(x - 1)}^{2} {(x + 3)}^{2}}{x - 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.3.3

Apply the distributive property.

$x + 1 = A (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) + \frac{(B) {(x - 1)}^{2} {(x + 3)}^{2}}{x - 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.4

Simplify and combine like terms.

Tap for more steps...

Step 1.1.9.4.1

Simplify each term.

Tap for more steps...

Step 1.1.9.4.1.1

Multiply $x$ by $x$ .

$x + 1 = A (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) + \frac{(B) {(x - 1)}^{2} {(x + 3)}^{2}}{x - 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.4.1.2

Move $3$ to the left of $x$ .

$x + 1 = A (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) + \frac{(B) {(x - 1)}^{2} {(x + 3)}^{2}}{x - 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.4.1.3

Multiply $3$ by $3$ .

$x + 1 = A (x^{2} + 3 x + 3 x + 9) + \frac{(B) {(x - 1)}^{2} {(x + 3)}^{2}}{x - 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.4.2

Add $3 x$ and $3 x$ .

$x + 1 = A (x^{2} + 6 x + 9) + \frac{(B) {(x - 1)}^{2} {(x + 3)}^{2}}{x - 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.5

Apply the distributive property.

$x + 1 = A x^{2} + A (6 x) + A \cdot 9 + \frac{(B) {(x - 1)}^{2} {(x + 3)}^{2}}{x - 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.6

Simplify.

Tap for more steps...

Step 1.1.9.6.1

Rewrite using the commutative property of multiplication.

$x + 1 = A x^{2} + 6 A x + A \cdot 9 + \frac{(B) {(x - 1)}^{2} {(x + 3)}^{2}}{x - 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.6.2

Move $9$ to the left of $A$ .

$x + 1 = A x^{2} + 6 A x + 9 \cdot A + \frac{(B) {(x - 1)}^{2} {(x + 3)}^{2}}{x - 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.7

Cancel the common factor of ${(x - 1)}^{2}$ and $x - 1$ .

Tap for more steps...

Step 1.1.9.7.1

Factor $x - 1$ out of $(B) {(x - 1)}^{2} {(x + 3)}^{2}$ .

$x + 1 = A x^{2} + 6 A x + 9 A + \frac{(x - 1) (B (x - 1) {(x + 3)}^{2})}{x - 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.7.2

Cancel the common factors.

Tap for more steps...

Step 1.1.9.7.2.1

Multiply by $1$ .

$x + 1 = A x^{2} + 6 A x + 9 A + \frac{(x - 1) (B (x - 1) {(x + 3)}^{2})}{(x - 1) \cdot 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.7.2.2

Cancel the common factor.

$x + 1 = A x^{2} + 6 A x + 9 A + \frac{(x - 1) (B (x - 1) {(x + 3)}^{2})}{(x - 1) \cdot 1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.7.2.3

Rewrite the expression.

$x + 1 = A x^{2} + 6 A x + 9 A + \frac{B (x - 1) {(x + 3)}^{2}}{1} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.7.2.4

Divide $B (x - 1) {(x + 3)}^{2}$ by $1$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B (x - 1) {(x + 3)}^{2} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.8

Apply the distributive property.

$x + 1 = A x^{2} + 6 A x + 9 A + (B x + B \cdot - 1) {(x + 3)}^{2} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.9

Move $- 1$ to the left of $B$ .

$x + 1 = A x^{2} + 6 A x + 9 A + (B x - 1 \cdot B) {(x + 3)}^{2} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.10

Rewrite $- 1 B$ as $- B$ .

$x + 1 = A x^{2} + 6 A x + 9 A + (B x - B) {(x + 3)}^{2} + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.11

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$x + 1 = A x^{2} + 6 A x + 9 A + (B x - B) ((x + 3) (x + 3)) + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.12

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Tap for more steps...

Step 1.1.9.12.1

Apply the distributive property.

$x + 1 = A x^{2} + 6 A x + 9 A + (B x - B) (x (x + 3) + 3 (x + 3)) + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.12.2

Apply the distributive property.

$x + 1 = A x^{2} + 6 A x + 9 A + (B x - B) (x \cdot x + x \cdot 3 + 3 (x + 3)) + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.12.3

Apply the distributive property.

$x + 1 = A x^{2} + 6 A x + 9 A + (B x - B) (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.13

Simplify and combine like terms.

Tap for more steps...

Step 1.1.9.13.1

Simplify each term.

Tap for more steps...

Step 1.1.9.13.1.1

Multiply $x$ by $x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + (B x - B) (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.13.1.2

Move $3$ to the left of $x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + (B x - B) (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.13.1.3

Multiply $3$ by $3$ .

$x + 1 = A x^{2} + 6 A x + 9 A + (B x - B) (x^{2} + 3 x + 3 x + 9) + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.13.2

Add $3 x$ and $3 x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + (B x - B) (x^{2} + 6 x + 9) + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.14

Expand $(B x - B) (x^{2} + 6 x + 9)$ by multiplying each term in the first expression by each term in the second expression.

$x + 1 = A x^{2} + 6 A x + 9 A + B x \cdot x^{2} + B x (6 x) + B x \cdot 9 - B x^{2} - B (6 x) - B \cdot 9 + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.15

Simplify each term.

Tap for more steps...

Step 1.1.9.15.1

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 1.1.9.15.1.1

Move $x^{2}$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B (x^{2} x) + B x (6 x) + B x \cdot 9 - B x^{2} - B (6 x) - B \cdot 9 + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.15.1.2

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 1.1.9.15.1.2.1

Raise $x$ to the power of $1$ .

Step 1.1.9.15.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{2 + 1} + B x (6 x) + B x \cdot 9 - B x^{2} - B (6 x) - B \cdot 9 + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.15.1.3

Add $2$ and $1$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + B x (6 x) + B x \cdot 9 - B x^{2} - B (6 x) - B \cdot 9 + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.15.2

Rewrite using the commutative property of multiplication.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 6 B x \cdot x + B x \cdot 9 - B x^{2} - B (6 x) - B \cdot 9 + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.15.3

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.1.9.15.3.1

Move $x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 6 B (x \cdot x) + B x \cdot 9 - B x^{2} - B (6 x) - B \cdot 9 + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.15.3.2

Multiply $x$ by $x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 6 B x^{2} + B x \cdot 9 - B x^{2} - B (6 x) - B \cdot 9 + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.15.4

Move $9$ to the left of $B x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 6 B x^{2} + 9 \cdot (B x) - B x^{2} - B (6 x) - B \cdot 9 + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.15.5

Rewrite using the commutative property of multiplication.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 6 B x^{2} + 9 B x - B x^{2} - 1 \cdot (6 B x) - B \cdot 9 + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.15.6

Multiply $- 1$ by $6$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 6 B x^{2} + 9 B x - B x^{2} - 6 B x - B \cdot 9 + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.15.7

Multiply $9$ by $- 1$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 6 B x^{2} + 9 B x - B x^{2} - 6 B x - 9 B + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.16

Subtract $B x^{2}$ from $6 B x^{2}$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 9 B x - 6 B x - 9 B + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.17

Subtract $6 B x$ from $9 B x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.18

Cancel the common factor of ${(x + 3)}^{2}$ .

Tap for more steps...

Step 1.1.9.18.1

Cancel the common factor.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + \frac{(C) {(x - 1)}^{2} {(x + 3)}^{2}}{{(x + 3)}^{2}} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.18.2

Divide $(C) {(x - 1)}^{2}$ by $1$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + (C) {(x - 1)}^{2} + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.19

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C ((x - 1) (x - 1)) + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.20

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Tap for more steps...

Step 1.1.9.20.1

Apply the distributive property.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C (x (x - 1) - 1 (x - 1)) + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.20.2

Apply the distributive property.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C (x \cdot x + x \cdot - 1 - 1 (x - 1)) + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.20.3

Apply the distributive property.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.21

Simplify and combine like terms.

Tap for more steps...

Step 1.1.9.21.1

Simplify each term.

Tap for more steps...

Step 1.1.9.21.1.1

Multiply $x$ by $x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.21.1.2

Move $- 1$ to the left of $x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.21.1.3

Rewrite $- 1 x$ as $- x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C (x^{2} - x - 1 x - 1 \cdot - 1) + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.21.1.4

Rewrite $- 1 x$ as $- x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C (x^{2} - x - x - 1 \cdot - 1) + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.21.1.5

Multiply $- 1$ by $- 1$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C (x^{2} - x - x + 1) + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.21.2

Subtract $x$ from $- x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C (x^{2} - 2 x + 1) + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.22

Apply the distributive property.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} + C (- 2 x) + C \cdot 1 + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.23

Simplify.

Tap for more steps...

Step 1.1.9.23.1

Rewrite using the commutative property of multiplication.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C \cdot 1 + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.23.2

Multiply $C$ by $1$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + \frac{(D) {(x - 1)}^{2} {(x + 3)}^{2}}{x + 3}$

Step 1.1.9.24

Cancel the common factor of ${(x + 3)}^{2}$ and $x + 3$ .

Tap for more steps...

Step 1.1.9.24.1

Factor $x + 3$ out of $(D) {(x - 1)}^{2} {(x + 3)}^{2}$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + \frac{(x + 3) ((D) {(x - 1)}^{2} (x + 3))}{x + 3}$

Step 1.1.9.24.2

Cancel the common factors.

Tap for more steps...

Step 1.1.9.24.2.1

Multiply by $1$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + \frac{(x + 3) ((D) {(x - 1)}^{2} (x + 3))}{(x + 3) \cdot 1}$

Step 1.1.9.24.2.2

Cancel the common factor.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + \frac{(x + 3) ((D) {(x - 1)}^{2} (x + 3))}{(x + 3) \cdot 1}$

Step 1.1.9.24.2.3

Rewrite the expression.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + \frac{(D) {(x - 1)}^{2} (x + 3)}{1}$

Step 1.1.9.24.2.4

Divide $(D) {(x - 1)}^{2} (x + 3)$ by $1$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + (D) {(x - 1)}^{2} (x + 3)$

Step 1.1.9.25

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D ((x - 1) (x - 1)) (x + 3)$

Step 1.1.9.26

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Tap for more steps...

Step 1.1.9.26.1

Apply the distributive property.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D (x (x - 1) - 1 (x - 1)) (x + 3)$

Step 1.1.9.26.2

Apply the distributive property.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D (x \cdot x + x \cdot - 1 - 1 (x - 1)) (x + 3)$

Step 1.1.9.26.3

Apply the distributive property.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) (x + 3)$

Step 1.1.9.27

Simplify and combine like terms.

Tap for more steps...

Step 1.1.9.27.1

Simplify each term.

Tap for more steps...

Step 1.1.9.27.1.1

Multiply $x$ by $x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) (x + 3)$

Step 1.1.9.27.1.2

Move $- 1$ to the left of $x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) (x + 3)$

Step 1.1.9.27.1.3

Rewrite $- 1 x$ as $- x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D (x^{2} - x - 1 x - 1 \cdot - 1) (x + 3)$

Step 1.1.9.27.1.4

Rewrite $- 1 x$ as $- x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D (x^{2} - x - x - 1 \cdot - 1) (x + 3)$

Step 1.1.9.27.1.5

Multiply $- 1$ by $- 1$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D (x^{2} - x - x + 1) (x + 3)$

Step 1.1.9.27.2

Subtract $x$ from $- x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D (x^{2} - 2 x + 1) (x + 3)$

Step 1.1.9.28

Apply the distributive property.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + (D x^{2} + D (- 2 x) + D \cdot 1) (x + 3)$

Step 1.1.9.29

Simplify.

Tap for more steps...

Step 1.1.9.29.1

Rewrite using the commutative property of multiplication.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + (D x^{2} - 2 D x + D \cdot 1) (x + 3)$

Step 1.1.9.29.2

Multiply $D$ by $1$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + (D x^{2} - 2 D x + D) (x + 3)$

Step 1.1.9.30

Expand $(D x^{2} - 2 D x + D) (x + 3)$ by multiplying each term in the first expression by each term in the second expression.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D x^{2} x + D x^{2} \cdot 3 - 2 D x \cdot x - 2 D x \cdot 3 + D x + D \cdot 3$

Step 1.1.9.31

Simplify each term.

Tap for more steps...

Step 1.1.9.31.1

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.1.9.31.1.1

Move $x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D (x \cdot x^{2}) + D x^{2} \cdot 3 - 2 D x \cdot x - 2 D x \cdot 3 + D x + D \cdot 3$

Step 1.1.9.31.1.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 1.1.9.31.1.2.1

Raise $x$ to the power of $1$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D (x \cdot x^{2}) + D x^{2} \cdot 3 - 2 D x \cdot x - 2 D x \cdot 3 + D x + D \cdot 3$

Step 1.1.9.31.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D x^{1 + 2} + D x^{2} \cdot 3 - 2 D x \cdot x - 2 D x \cdot 3 + D x + D \cdot 3$

Step 1.1.9.31.1.3

Add $1$ and $2$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D x^{3} + D x^{2} \cdot 3 - 2 D x \cdot x - 2 D x \cdot 3 + D x + D \cdot 3$

Step 1.1.9.31.2

Move $3$ to the left of $D x^{2}$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D x^{3} + 3 \cdot (D x^{2}) - 2 D x \cdot x - 2 D x \cdot 3 + D x + D \cdot 3$

Step 1.1.9.31.3

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.1.9.31.3.1

Move $x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D x^{3} + 3 D x^{2} - 2 D (x \cdot x) - 2 D x \cdot 3 + D x + D \cdot 3$

Step 1.1.9.31.3.2

Multiply $x$ by $x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D x^{3} + 3 D x^{2} - 2 D x^{2} - 2 D x \cdot 3 + D x + D \cdot 3$

Step 1.1.9.31.4

Multiply $3$ by $- 2$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D x^{3} + 3 D x^{2} - 2 D x^{2} - 6 D x + D x + D \cdot 3$

Step 1.1.9.31.5

Move $3$ to the left of $D$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D x^{3} + 3 D x^{2} - 2 D x^{2} - 6 D x + D x + 3 D$

Step 1.1.9.32

Subtract $2 D x^{2}$ from $3 D x^{2}$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D x^{3} + 1 D x^{2} - 6 D x + D x + 3 D$

Step 1.1.9.33

Multiply $D$ by $1$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D x^{3} + D x^{2} - 6 D x + D x + 3 D$

Step 1.1.9.34

Add $- 6 D x$ and $D x$ .

$x + 1 = A x^{2} + 6 A x + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D x^{3} + D x^{2} - 5 D x + 3 D$

Step 1.1.10

Simplify the expression.

Tap for more steps...

Step 1.1.10.1

Move $A$ .

$x + 1 = A x^{2} + 6 x A + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D x^{3} + D x^{2} - 5 D x + 3 D$

Step 1.1.10.2

Reorder $B$ and $x^{3}$ .

$x + 1 = A x^{2} + 6 x A + 9 A + B x^{3} + 5 B x^{2} + 3 B x - 9 B + C x^{2} - 2 C x + C + D x^{3} + D x^{2} - 5 D x + 3 D$

Step 1.1.10.3

Move $B$ .

$x + 1 = A x^{2} + 6 x A + 9 A + B x^{3} + 5 x^{2} B + 3 B x - 9 B + C x^{2} - 2 C x + C + D x^{3} + D x^{2} - 5 D x + 3 D$

Step 1.1.10.4

Move $B$ .

$x + 1 = A x^{2} + 6 x A + 9 A + B x^{3} + 5 x^{2} B + 3 x B - 9 B + C x^{2} - 2 C x + C + D x^{3} + D x^{2} - 5 D x + 3 D$

Step 1.1.10.5

Move $C$ .

$x + 1 = A x^{2} + 6 x A + 9 A + B x^{3} + 5 x^{2} B + 3 x B - 9 B + C x^{2} - 2 C x + D x^{3} + D x^{2} - 5 D x + C + 3 D$

Step 1.1.10.6

Move $- 9 B$ .

$x + 1 = A x^{2} + 6 x A + 9 A + B x^{3} + 5 x^{2} B + 3 x B + C x^{2} - 2 C x + D x^{3} + D x^{2} - 5 D x - 9 B + C + 3 D$

Step 1.1.10.7

Move $9 A$ .

$x + 1 = A x^{2} + 6 x A + B x^{3} + 5 x^{2} B + 3 x B + C x^{2} - 2 C x + D x^{3} + D x^{2} - 5 D x + 9 A - 9 B + C + 3 D$

Step 1.1.10.8

Move $- 2 C x$ .

$x + 1 = A x^{2} + 6 x A + B x^{3} + 5 x^{2} B + 3 x B + C x^{2} + D x^{3} + D x^{2} - 2 C x - 5 D x + 9 A - 9 B + C + 3 D$

Step 1.1.10.9

Move $3 x B$ .

$x + 1 = A x^{2} + 6 x A + B x^{3} + 5 x^{2} B + C x^{2} + D x^{3} + D x^{2} + 3 x B - 2 C x - 5 D x + 9 A - 9 B + C + 3 D$

Step 1.1.10.10

Move $6 x A$ .

$x + 1 = A x^{2} + B x^{3} + 5 x^{2} B + C x^{2} + D x^{3} + D x^{2} + 6 x A + 3 x B - 2 C x - 5 D x + 9 A - 9 B + C + 3 D$

Step 1.1.10.11

Move $C x^{2}$ .

$x + 1 = A x^{2} + B x^{3} + 5 x^{2} B + D x^{3} + C x^{2} + D x^{2} + 6 x A + 3 x B - 2 C x - 5 D x + 9 A - 9 B + C + 3 D$

Step 1.1.10.12

Move $5 x^{2} B$ .

$x + 1 = A x^{2} + B x^{3} + D x^{3} + 5 x^{2} B + C x^{2} + D x^{2} + 6 x A + 3 x B - 2 C x - 5 D x + 9 A - 9 B + C + 3 D$

Step 1.1.10.13

Move $A x^{2}$ .

$x + 1 = B x^{3} + D x^{3} + A x^{2} + 5 x^{2} B + C x^{2} + D x^{2} + 6 x A + 3 x B - 2 C x - 5 D x + 9 A - 9 B + C + 3 D$

Step 1.2

Create equations for the partial fraction variables and use them to set up a system of equations.

Tap for more steps...

Step 1.2.1

Create an equation for the partial fraction variables by equating the coefficients of $x^{3}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = B + D$

Step 1.2.2

Create an equation for the partial fraction variables by equating the coefficients of $x^{2}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = A + 5 B + C + D$

Step 1.2.3

Create an equation for the partial fraction variables by equating the coefficients of $x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = 6 A + 3 B - 2 C - 5 D$

Step 1.2.4

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $x$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = 9 A - 9 B + C + 3 D$

Step 1.2.5

Set up the system of equations to find the coefficients of the partial fractions.

$0 = B + D$

$0 = A + 5 B + C + D$

$1 = 6 A + 3 B - 2 C - 5 D$

$1 = 9 A - 9 B + C + 3 D$

$0 = B + D$

$0 = A + 5 B + C + D$

$1 = 6 A + 3 B - 2 C - 5 D$

$1 = 9 A - 9 B + C + 3 D$

Step 1.3

Solve the system of equations.

Tap for more steps...

Step 1.3.1

Solve for $B$ in $0 = B + D$ .

Tap for more steps...

Step 1.3.1.1

Rewrite the equation as $B + D = 0$ .

$B + D = 0$

$0 = A + 5 B + C + D$

$1 = 6 A + 3 B - 2 C - 5 D$

$1 = 9 A - 9 B + C + 3 D$

Step 1.3.1.2

Subtract $D$ from both sides of the equation.

$B = - D$

$0 = A + 5 B + C + D$

$1 = 6 A + 3 B - 2 C - 5 D$

$1 = 9 A - 9 B + C + 3 D$

$B = - D$

$0 = A + 5 B + C + D$

$1 = 6 A + 3 B - 2 C - 5 D$

$1 = 9 A - 9 B + C + 3 D$

Step 1.3.2

Replace all occurrences of $B$ with $- D$ in each equation.

Tap for more steps...

Step 1.3.2.1

Replace all occurrences of $B$ in $0 = A + 5 B + C + D$ with $- D$ .

$0 = A + 5 (- D) + C + D$

$B = - D$

$1 = 6 A + 3 B - 2 C - 5 D$

$1 = 9 A - 9 B + C + 3 D$

Step 1.3.2.2

Simplify the right side.

Tap for more steps...

Step 1.3.2.2.1

Simplify $A + 5 (- D) + C + D$ .

Tap for more steps...

Step 1.3.2.2.1.1

Multiply $- 1$ by $5$ .

$0 = A - 5 D + C + D$

$B = - D$

$1 = 6 A + 3 B - 2 C - 5 D$

$1 = 9 A - 9 B + C + 3 D$

Step 1.3.2.2.1.2

Add $- 5 D$ and $D$ .

$0 = A + C - 4 D$

$B = - D$

$1 = 6 A + 3 B - 2 C - 5 D$

$1 = 9 A - 9 B + C + 3 D$

$0 = A + C - 4 D$

$B = - D$

$1 = 6 A + 3 B - 2 C - 5 D$

$1 = 9 A - 9 B + C + 3 D$

$0 = A + C - 4 D$

$B = - D$

$1 = 6 A + 3 B - 2 C - 5 D$

$1 = 9 A - 9 B + C + 3 D$

Step 1.3.2.3

Replace all occurrences of $B$ in $1 = 6 A + 3 B - 2 C - 5 D$ with $- D$ .

$1 = 6 A + 3 (- D) - 2 C - 5 D$

$0 = A + C - 4 D$

$B = - D$

$1 = 9 A - 9 B + C + 3 D$

Step 1.3.2.4

Simplify the right side.

Tap for more steps...

Step 1.3.2.4.1

Simplify $6 A + 3 (- D) - 2 C - 5 D$ .

Tap for more steps...

Step 1.3.2.4.1.1

Multiply $- 1$ by $3$ .

$1 = 6 A - 3 D - 2 C - 5 D$

$0 = A + C - 4 D$

$B = - D$

$1 = 9 A - 9 B + C + 3 D$

Step 1.3.2.4.1.2

Subtract $5 D$ from $- 3 D$ .

$1 = 6 A - 2 C - 8 D$

$0 = A + C - 4 D$

$B = - D$

$1 = 9 A - 9 B + C + 3 D$

$1 = 6 A - 2 C - 8 D$

$0 = A + C - 4 D$

$B = - D$

$1 = 9 A - 9 B + C + 3 D$

$1 = 6 A - 2 C - 8 D$

$0 = A + C - 4 D$

$B = - D$

$1 = 9 A - 9 B + C + 3 D$

Step 1.3.2.5

Replace all occurrences of $B$ in $1 = 9 A - 9 B + C + 3 D$ with $- D$ .

$1 = 9 A - 9 (- D) + C + 3 D$

$1 = 6 A - 2 C - 8 D$

$0 = A + C - 4 D$

$B = - D$

Step 1.3.2.6

Simplify the right side.

Tap for more steps...

Step 1.3.2.6.1

Simplify $9 A - 9 (- D) + C + 3 D$ .

Tap for more steps...

Step 1.3.2.6.1.1

Multiply $- 1$ by $- 9$ .

$1 = 9 A + 9 D + C + 3 D$

$1 = 6 A - 2 C - 8 D$

$0 = A + C - 4 D$

$B = - D$

Step 1.3.2.6.1.2

Add $9 D$ and $3 D$ .

$1 = 9 A + C + 12 D$

$1 = 6 A - 2 C - 8 D$

$0 = A + C - 4 D$

$B = - D$

$1 = 9 A + C + 12 D$

$1 = 6 A - 2 C - 8 D$

$0 = A + C - 4 D$

$B = - D$

$1 = 9 A + C + 12 D$

$1 = 6 A - 2 C - 8 D$

$0 = A + C - 4 D$

$B = - D$

$1 = 9 A + C + 12 D$

$1 = 6 A - 2 C - 8 D$

$0 = A + C - 4 D$

$B = - D$

Step 1.3.3

Solve for $C$ in $1 = 9 A + C + 12 D$ .

Tap for more steps...

Step 1.3.3.1

Rewrite the equation as $9 A + C + 12 D = 1$ .

$9 A + C + 12 D = 1$

$1 = 6 A - 2 C - 8 D$

$0 = A + C - 4 D$

$B = - D$

Step 1.3.3.2

Move all terms not containing $C$ to the right side of the equation.

Tap for more steps...

Step 1.3.3.2.1

Subtract $9 A$ from both sides of the equation.

$C + 12 D = 1 - 9 A$

$1 = 6 A - 2 C - 8 D$

$0 = A + C - 4 D$

$B = - D$

Step 1.3.3.2.2

Subtract $12 D$ from both sides of the equation.

$C = 1 - 9 A - 12 D$

$1 = 6 A - 2 C - 8 D$

$0 = A + C - 4 D$

$B = - D$

$C = 1 - 9 A - 12 D$

$1 = 6 A - 2 C - 8 D$

$0 = A + C - 4 D$

$B = - D$

$C = 1 - 9 A - 12 D$

$1 = 6 A - 2 C - 8 D$

$0 = A + C - 4 D$

$B = - D$

Step 1.3.4

Replace all occurrences of $C$ with $1 - 9 A - 12 D$ in each equation.

Tap for more steps...

Step 1.3.4.1

Replace all occurrences of $C$ in $1 = 6 A - 2 C - 8 D$ with $1 - 9 A - 12 D$ .

$1 = 6 A - 2 (1 - 9 A - 12 D) - 8 D$

$C = 1 - 9 A - 12 D$

$0 = A + C - 4 D$

$B = - D$

Step 1.3.4.2

Simplify the right side.

Tap for more steps...

Step 1.3.4.2.1

Simplify $6 A - 2 (1 - 9 A - 12 D) - 8 D$ .

Tap for more steps...

Step 1.3.4.2.1.1

Simplify each term.

Tap for more steps...

Step 1.3.4.2.1.1.1

Apply the distributive property.

$1 = 6 A - 2 \cdot 1 - 2 (- 9 A) - 2 (- 12 D) - 8 D$

$C = 1 - 9 A - 12 D$

$0 = A + C - 4 D$

$B = - D$

Step 1.3.4.2.1.1.2

Simplify.

Tap for more steps...

Step 1.3.4.2.1.1.2.1

Multiply $- 2$ by $1$ .

$1 = 6 A - 2 - 2 (- 9 A) - 2 (- 12 D) - 8 D$

$C = 1 - 9 A - 12 D$

$0 = A + C - 4 D$

$B = - D$

Step 1.3.4.2.1.1.2.2

Multiply $- 9$ by $- 2$ .

$1 = 6 A - 2 + 18 A - 2 (- 12 D) - 8 D$

$C = 1 - 9 A - 12 D$

$0 = A + C - 4 D$

$B = - D$

Step 1.3.4.2.1.1.2.3

Multiply $- 12$ by $- 2$ .

$1 = 6 A - 2 + 18 A + 24 D - 8 D$

$C = 1 - 9 A - 12 D$

$0 = A + C - 4 D$

$B = - D$

$1 = 6 A - 2 + 18 A + 24 D - 8 D$

$C = 1 - 9 A - 12 D$

$0 = A + C - 4 D$

$B = - D$

$1 = 6 A - 2 + 18 A + 24 D - 8 D$

$C = 1 - 9 A - 12 D$

$0 = A + C - 4 D$

$B = - D$

Step 1.3.4.2.1.2

Simplify by adding terms.

Tap for more steps...

Step 1.3.4.2.1.2.1

Add $6 A$ and $18 A$ .

$1 = 24 A - 2 + 24 D - 8 D$

$C = 1 - 9 A - 12 D$

$0 = A + C - 4 D$

$B = - D$

Step 1.3.4.2.1.2.2

Subtract $8 D$ from $24 D$ .

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$0 = A + C - 4 D$

$B = - D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$0 = A + C - 4 D$

$B = - D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$0 = A + C - 4 D$

$B = - D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$0 = A + C - 4 D$

$B = - D$

Step 1.3.4.3

Replace all occurrences of $C$ in $0 = A + C - 4 D$ with $1 - 9 A - 12 D$ .

$0 = A + 1 - 9 A - 12 D - 4 D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.4.4

Simplify $0 = A + 1 - 9 A - 12 D - 4 D$ .

Tap for more steps...

Step 1.3.4.4.1

Simplify the left side.

Tap for more steps...

Step 1.3.4.4.1.1

Remove parentheses.

$0 = A + 1 - 9 A - 12 D - 4 D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$0 = A + 1 - 9 A - 12 D - 4 D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.4.4.2

Simplify the right side.

Tap for more steps...

Step 1.3.4.4.2.1

Simplify $A + 1 - 9 A - 12 D - 4 D$ .

Tap for more steps...

Step 1.3.4.4.2.1.1

Subtract $9 A$ from $A$ .

$0 = - 8 A + 1 - 12 D - 4 D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.4.4.2.1.2

Subtract $4 D$ from $- 12 D$ .

$0 = - 8 A + 1 - 16 D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$0 = - 8 A + 1 - 16 D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$0 = - 8 A + 1 - 16 D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$0 = - 8 A + 1 - 16 D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$0 = - 8 A + 1 - 16 D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.5

Solve for $A$ in $0 = - 8 A + 1 - 16 D$ .

Tap for more steps...

Step 1.3.5.1

Rewrite the equation as $- 8 A + 1 - 16 D = 0$ .

$- 8 A + 1 - 16 D = 0$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.5.2

Move all terms not containing $A$ to the right side of the equation.

Tap for more steps...

Step 1.3.5.2.1

Subtract $1$ from both sides of the equation.

$- 8 A - 16 D = - 1$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.5.2.2

Add $16 D$ to both sides of the equation.

$- 8 A = - 1 + 16 D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$- 8 A = - 1 + 16 D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.5.3

Divide each term in $- 8 A = - 1 + 16 D$ by $- 8$ and simplify.

Tap for more steps...

Step 1.3.5.3.1

Divide each term in $- 8 A = - 1 + 16 D$ by $- 8$ .

$\frac{- 8 A}{- 8} = \frac{- 1}{- 8} + \frac{16 D}{- 8}$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.5.3.2

Simplify the left side.

Tap for more steps...

Step 1.3.5.3.2.1

Cancel the common factor of $- 8$ .

Tap for more steps...

Step 1.3.5.3.2.1.1

Cancel the common factor.

$\frac{- 8 A}{- 8} = \frac{- 1}{- 8} + \frac{16 D}{- 8}$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.5.3.2.1.2

Divide $A$ by $1$ .

$A = \frac{- 1}{- 8} + \frac{16 D}{- 8}$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$A = \frac{- 1}{- 8} + \frac{16 D}{- 8}$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$A = \frac{- 1}{- 8} + \frac{16 D}{- 8}$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.5.3.3

Simplify the right side.

Tap for more steps...

Step 1.3.5.3.3.1

Simplify each term.

Tap for more steps...

Step 1.3.5.3.3.1.1

Dividing two negative values results in a positive value.

$A = \frac{1}{8} + \frac{16 D}{- 8}$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.5.3.3.1.2

Cancel the common factor of $16$ and $- 8$ .

Tap for more steps...

Step 1.3.5.3.3.1.2.1

Factor $8$ out of $16 D$ .

$A = \frac{1}{8} + \frac{8 (2 D)}{- 8}$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.5.3.3.1.2.2

Move the negative one from the denominator of $\frac{2 D}{- 1}$ .

$A = \frac{1}{8} - 1 \cdot (2 D)$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$A = \frac{1}{8} - 1 \cdot (2 D)$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.5.3.3.1.3

Rewrite $- 1 \cdot (2 D)$ as $- (2 D)$ .

$A = \frac{1}{8} - (2 D)$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.5.3.3.1.4

Multiply $2$ by $- 1$ .

$A = \frac{1}{8} - 2 D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$A = \frac{1}{8} - 2 D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$A = \frac{1}{8} - 2 D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$A = \frac{1}{8} - 2 D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$A = \frac{1}{8} - 2 D$

$1 = 24 A - 2 + 16 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.6

Replace all occurrences of $A$ with $\frac{1}{8} - 2 D$ in each equation.

Tap for more steps...

Step 1.3.6.1

Replace all occurrences of $A$ in $1 = 24 A - 2 + 16 D$ with $\frac{1}{8} - 2 D$ .

$1 = 24 (\frac{1}{8} - 2 D) - 2 + 16 D$

$A = \frac{1}{8} - 2 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.6.2

Simplify the right side.

Tap for more steps...

Step 1.3.6.2.1

Simplify $24 (\frac{1}{8} - 2 D) - 2 + 16 D$ .

Tap for more steps...

Step 1.3.6.2.1.1

Simplify each term.

Tap for more steps...

Step 1.3.6.2.1.1.1

Apply the distributive property.

$1 = 24 (\frac{1}{8}) + 24 (- 2 D) - 2 + 16 D$

$A = \frac{1}{8} - 2 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.6.2.1.1.2

Cancel the common factor of $8$ .

Tap for more steps...

Step 1.3.6.2.1.1.2.1

Factor $8$ out of $24$ .

$1 = 8 (3) (\frac{1}{8}) + 24 (- 2 D) - 2 + 16 D$

$A = \frac{1}{8} - 2 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.6.2.1.1.2.2

Cancel the common factor.

$1 = 8 \cdot (3 (\frac{1}{8})) + 24 (- 2 D) - 2 + 16 D$

$A = \frac{1}{8} - 2 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.6.2.1.1.2.3

Rewrite the expression.

$1 = 3 + 24 (- 2 D) - 2 + 16 D$

$A = \frac{1}{8} - 2 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$1 = 3 + 24 (- 2 D) - 2 + 16 D$

$A = \frac{1}{8} - 2 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.6.2.1.1.3

Multiply $- 2$ by $24$ .

$1 = 3 - 48 D - 2 + 16 D$

$A = \frac{1}{8} - 2 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$1 = 3 - 48 D - 2 + 16 D$

$A = \frac{1}{8} - 2 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.6.2.1.2

Simplify by adding terms.

Tap for more steps...

Step 1.3.6.2.1.2.1

Subtract $2$ from $3$ .

$1 = - 48 D + 1 + 16 D$

$A = \frac{1}{8} - 2 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.6.2.1.2.2

Add $- 48 D$ and $16 D$ .

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$C = 1 - 9 A - 12 D$

$B = - D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$C = 1 - 9 A - 12 D$

$B = - D$

Step 1.3.6.3

Replace all occurrences of $A$ in $C = 1 - 9 A - 12 D$ with $\frac{1}{8} - 2 D$ .

$C = 1 - 9 (\frac{1}{8} - 2 D) - 12 D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.6.4

Simplify the right side.

Tap for more steps...

Step 1.3.6.4.1

Simplify $1 - 9 (\frac{1}{8} - 2 D) - 12 D$ .

Tap for more steps...

Step 1.3.6.4.1.1

Simplify each term.

Tap for more steps...

Step 1.3.6.4.1.1.1

Apply the distributive property.

$C = 1 - 9 (\frac{1}{8}) - 9 (- 2 D) - 12 D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.6.4.1.1.2

Combine $- 9$ and $\frac{1}{8}$ .

$C = 1 + \frac{- 9}{8} - 9 (- 2 D) - 12 D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.6.4.1.1.3

Multiply $- 2$ by $- 9$ .

$C = 1 + \frac{- 9}{8} + 18 D - 12 D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.6.4.1.1.4

Move the negative in front of the fraction.

$C = 1 - \frac{9}{8} + 18 D - 12 D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$B = - D$

$C = 1 - \frac{9}{8} + 18 D - 12 D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.6.4.1.2

Simplify by adding terms.

Tap for more steps...

Step 1.3.6.4.1.2.1

Write $1$ as a fraction with a common denominator.

$C = \frac{8}{8} - \frac{9}{8} + 18 D - 12 D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.6.4.1.2.2

Simplify the expression.

Tap for more steps...

Step 1.3.6.4.1.2.2.1

Combine the numerators over the common denominator.

$C = \frac{8 - 9}{8} + 18 D - 12 D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.6.4.1.2.2.2

Subtract $9$ from $8$ .

$C = \frac{- 1}{8} + 18 D - 12 D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.6.4.1.2.2.3

Move the negative in front of the fraction.

$C = - \frac{1}{8} + 18 D - 12 D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$B = - D$

$C = - \frac{1}{8} + 18 D - 12 D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.6.4.1.2.3

Subtract $12 D$ from $18 D$ .

$C = - \frac{1}{8} + 6 D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$B = - D$

$C = - \frac{1}{8} + 6 D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$B = - D$

$C = - \frac{1}{8} + 6 D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$B = - D$

$C = - \frac{1}{8} + 6 D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$B = - D$

$C = - \frac{1}{8} + 6 D$

$1 = - 32 D + 1$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.7

Solve for $D$ in $1 = - 32 D + 1$ .

Tap for more steps...

Step 1.3.7.1

Rewrite the equation as $- 32 D + 1 = 1$ .

$- 32 D + 1 = 1$

$C = - \frac{1}{8} + 6 D$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.7.2

Move all terms not containing $D$ to the right side of the equation.

Tap for more steps...

Step 1.3.7.2.1

Subtract $1$ from both sides of the equation.

$- 32 D = 1 - 1$

$C = - \frac{1}{8} + 6 D$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.7.2.2

Subtract $1$ from $1$ .

$- 32 D = 0$

$C = - \frac{1}{8} + 6 D$

$A = \frac{1}{8} - 2 D$

$B = - D$

$- 32 D = 0$

$C = - \frac{1}{8} + 6 D$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.7.3

Divide each term in $- 32 D = 0$ by $- 32$ and simplify.

Tap for more steps...

Step 1.3.7.3.1

Divide each term in $- 32 D = 0$ by $- 32$ .

$\frac{- 32 D}{- 32} = \frac{0}{- 32}$

$C = - \frac{1}{8} + 6 D$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.7.3.2

Simplify the left side.

Tap for more steps...

Step 1.3.7.3.2.1

Cancel the common factor of $- 32$ .

Tap for more steps...

Step 1.3.7.3.2.1.1

Cancel the common factor.

$\frac{- 32 D}{- 32} = \frac{0}{- 32}$

$C = - \frac{1}{8} + 6 D$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.7.3.2.1.2

Divide $D$ by $1$ .

$D = \frac{0}{- 32}$

$C = - \frac{1}{8} + 6 D$

$A = \frac{1}{8} - 2 D$

$B = - D$

$D = \frac{0}{- 32}$

$C = - \frac{1}{8} + 6 D$

$A = \frac{1}{8} - 2 D$

$B = - D$

$D = \frac{0}{- 32}$

$C = - \frac{1}{8} + 6 D$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.7.3.3

Simplify the right side.

Tap for more steps...

Step 1.3.7.3.3.1

Divide $0$ by $- 32$ .

$D = 0$

$C = - \frac{1}{8} + 6 D$

$A = \frac{1}{8} - 2 D$

$B = - D$

$D = 0$

$C = - \frac{1}{8} + 6 D$

$A = \frac{1}{8} - 2 D$

$B = - D$

$D = 0$

$C = - \frac{1}{8} + 6 D$

$A = \frac{1}{8} - 2 D$

$B = - D$

$D = 0$

$C = - \frac{1}{8} + 6 D$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.8

Replace all occurrences of $D$ with $0$ in each equation.

Tap for more steps...

Step 1.3.8.1

Replace all occurrences of $D$ in $C = - \frac{1}{8} + 6 D$ with $0$ .

$C = - \frac{1}{8} + 6 (0)$

$D = 0$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.8.2

Simplify the right side.

Tap for more steps...

Step 1.3.8.2.1

Simplify $- \frac{1}{8} + 6 (0)$ .

Tap for more steps...

Step 1.3.8.2.1.1

Multiply $6$ by $0$ .

$C = - \frac{1}{8} + 0$

$D = 0$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.8.2.1.2

Add $- \frac{1}{8}$ and $0$ .

$C = - \frac{1}{8}$

$D = 0$

$A = \frac{1}{8} - 2 D$

$B = - D$

$C = - \frac{1}{8}$

$D = 0$

$A = \frac{1}{8} - 2 D$

$B = - D$

$C = - \frac{1}{8}$

$D = 0$

$A = \frac{1}{8} - 2 D$

$B = - D$

Step 1.3.8.3

Replace all occurrences of $D$ in $A = \frac{1}{8} - 2 D$ with $0$ .

$A = \frac{1}{8} - 2 \cdot 0$

$C = - \frac{1}{8}$

$D = 0$

$B = - D$

Step 1.3.8.4

Simplify the right side.

Tap for more steps...

Step 1.3.8.4.1

Simplify $\frac{1}{8} - 2 \cdot 0$ .

Tap for more steps...

Step 1.3.8.4.1.1

Multiply $- 2$ by $0$ .

$A = \frac{1}{8} + 0$

$C = - \frac{1}{8}$

$D = 0$

$B = - D$

Step 1.3.8.4.1.2

Add $\frac{1}{8}$ and $0$ .

$A = \frac{1}{8}$

$C = - \frac{1}{8}$

$D = 0$

$B = - D$

$A = \frac{1}{8}$

$C = - \frac{1}{8}$

$D = 0$

$B = - D$

$A = \frac{1}{8}$

$C = - \frac{1}{8}$

$D = 0$

$B = - D$

Step 1.3.8.5

Replace all occurrences of $D$ in $B = - D$ with $0$ .

$B = - (0)$

$A = \frac{1}{8}$

$C = - \frac{1}{8}$

$D = 0$

Step 1.3.8.6

Simplify the right side.

Tap for more steps...

Step 1.3.8.6.1

Multiply $- 1$ by $0$ .

$B = 0$

$A = \frac{1}{8}$

$C = - \frac{1}{8}$

$D = 0$

$B = 0$

$A = \frac{1}{8}$

$C = - \frac{1}{8}$

$D = 0$

$B = 0$

$A = \frac{1}{8}$

$C = - \frac{1}{8}$

$D = 0$

Step 1.3.9

List all of the solutions.

$B = 0, A = \frac{1}{8}, C = - \frac{1}{8}, D = 0$

Step 1.4

Replace each of the partial fraction coefficients in $\frac{A}{{(x - 1)}^{2}} + \frac{B}{x - 1} + \frac{C}{{(x + 3)}^{2}} + \frac{D}{x + 3}$ with the values found for $A$ , $B$ , $C$ , and $D$ .

$\frac{\frac{1}{8}}{{(x - 1)}^{2}} + \frac{0}{x - 1} + \frac{- \frac{1}{8}}{{(x + 3)}^{2}} + \frac{0}{x + 3}$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{8} \cdot \frac{1}{{(x - 1)}^{2}} + \frac{0}{x - 1} + \frac{- \frac{1}{8}}{{(x + 3)}^{2}} + \frac{0}{x + 3}$

Step 1.5.2

Combine.

$\frac{1 \cdot 1}{8 {(x - 1)}^{2}} + \frac{0}{x - 1} + \frac{- \frac{1}{8}}{{(x + 3)}^{2}} + \frac{0}{x + 3}$

Step 1.5.3

Multiply $1$ by $1$ .

$\frac{1}{8 {(x - 1)}^{2}} + \frac{0}{x - 1} + \frac{- \frac{1}{8}}{{(x + 3)}^{2}} + \frac{0}{x + 3}$

Step 1.5.4

Divide $0$ by $x - 1$ .

$\frac{1}{8 {(x - 1)}^{2}} + 0 + \frac{- \frac{1}{8}}{{(x + 3)}^{2}} + \frac{0}{x + 3}$

Step 1.5.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{8 {(x - 1)}^{2}} + 0 - \frac{1}{8} \cdot \frac{1}{{(x + 3)}^{2}} + \frac{0}{x + 3}$

Step 1.5.6

Multiply $\frac{1}{{(x + 3)}^{2}}$ by $\frac{1}{8}$ .

$\frac{1}{8 {(x - 1)}^{2}} + 0 - \frac{1}{{(x + 3)}^{2} \cdot 8} + \frac{0}{x + 3}$

Step 1.5.7

Move $8$ to the left of ${(x + 3)}^{2}$ .

$\frac{1}{8 {(x - 1)}^{2}} + 0 - \frac{1}{8 {(x + 3)}^{2}} + \frac{0}{x + 3}$

Step 1.5.8

Divide $0$ by $x + 3$ .

$\frac{1}{8 {(x - 1)}^{2}} + 0 - \frac{1}{8 {(x + 3)}^{2}} + 0$

Step 1.5.9

Remove the zero from the expression.

$\int \frac{1}{8 {(x - 1)}^{2}} - \frac{1}{8 {(x + 3)}^{2}} d x$

Step 2

Split the single integral into multiple integrals.

$\int \frac{1}{8 {(x - 1)}^{2}} d x + \int - \frac{1}{8 {(x + 3)}^{2}} d x$

Step 3

Since $\frac{1}{8}$ is constant with respect to $x$ , move $\frac{1}{8}$ out of the integral.

$\frac{1}{8} \int \frac{1}{{(x - 1)}^{2}} d x + \int - \frac{1}{8 {(x + 3)}^{2}} d x$

Step 4

Let $u_{1} = x - 1$ . Then $d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 4.1

Let $u_{1} = x - 1$ . Find $\frac{d u_{1}}{d x}$ .

Tap for more steps...

Step 4.1.1

Differentiate $x - 1$ .

$\frac{d}{d x} [x - 1]$

Step 4.1.2

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- 1]$

Step 4.1.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$1 + 0$

Step 4.1.5

Add $1$ and $0$ .

$1$

Step 4.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\frac{1}{8} \int \frac{1}{{u_{1}}^{2}} d u_{1} + \int - \frac{1}{8 {(x + 3)}^{2}} d x$

Step 5

Apply basic rules of exponents.

Tap for more steps...

Step 5.1

Move ${u_{1}}^{2}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{8} \int {({u_{1}}^{2})}^{- 1} d u_{1} + \int - \frac{1}{8 {(x + 3)}^{2}} d x$

Step 5.2

Multiply the exponents in ${({u_{1}}^{2})}^{- 1}$ .

Tap for more steps...

Step 5.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{8} \int {u_{1}}^{2 \cdot - 1} d u_{1} + \int - \frac{1}{8 {(x + 3)}^{2}} d x$

Step 5.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{8} \int {u_{1}}^{- 2} d u_{1} + \int - \frac{1}{8 {(x + 3)}^{2}} d x$

Step 6

By the Power Rule, the integral of ${u_{1}}^{- 2}$ with respect to $u_{1}$ is $- {u_{1}}^{- 1}$ .

$\frac{1}{8} (- {u_{1}}^{- 1} + C) + \int - \frac{1}{8 {(x + 3)}^{2}} d x$

Step 7

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{8} (- {u_{1}}^{- 1} + C) - \int \frac{1}{8 {(x + 3)}^{2}} d x$

Step 8

Since $\frac{1}{8}$ is constant with respect to $x$ , move $\frac{1}{8}$ out of the integral.

$\frac{1}{8} (- {u_{1}}^{- 1} + C) - (\frac{1}{8} \int \frac{1}{{(x + 3)}^{2}} d x)$

Step 9

Let $u_{2} = x + 3$ . Then $d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 9.1

Let $u_{2} = x + 3$ . Find $\frac{d u_{2}}{d x}$ .

Tap for more steps...

Step 9.1.1

Differentiate $x + 3$ .

$\frac{d}{d x} [x + 3]$

Step 9.1.2

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 9.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [3]$

Step 9.1.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$1 + 0$

Step 9.1.5

Add $1$ and $0$ .

$1$

Step 9.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{8} (- {u_{1}}^{- 1} + C) - \frac{1}{8} \int \frac{1}{{u_{2}}^{2}} d u_{2}$

Step 10

Apply basic rules of exponents.

Tap for more steps...

Step 10.1

Move ${u_{2}}^{2}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{8} (- {u_{1}}^{- 1} + C) - \frac{1}{8} \int {({u_{2}}^{2})}^{- 1} d u_{2}$

Step 10.2

Multiply the exponents in ${({u_{2}}^{2})}^{- 1}$ .

Tap for more steps...

Step 10.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{8} (- {u_{1}}^{- 1} + C) - \frac{1}{8} \int {u_{2}}^{2 \cdot - 1} d u_{2}$

Step 10.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{8} (- {u_{1}}^{- 1} + C) - \frac{1}{8} \int {u_{2}}^{- 2} d u_{2}$

Step 11

By the Power Rule, the integral of ${u_{2}}^{- 2}$ with respect to $u_{2}$ is $- {u_{2}}^{- 1}$ .

$\frac{1}{8} (- {u_{1}}^{- 1} + C) - \frac{1}{8} (- {u_{2}}^{- 1} + C)$

Step 12

Simplify.

Tap for more steps...

Step 12.1

Simplify.

$\frac{1}{8} (- \frac{1}{u_{1}}) - \frac{1}{8} (- {u_{2}}^{- 1}) + C$

Step 12.2

Simplify.

Tap for more steps...

Step 12.2.1

Multiply $\frac{1}{8}$ by $\frac{1}{u_{1}}$ .

$- \frac{1}{8 u_{1}} - \frac{1}{8} (- {u_{2}}^{- 1}) + C$

Step 12.2.2

Multiply $- 1$ by $- 1$ .

$- \frac{1}{8 u_{1}} + 1 (\frac{1}{8}) {u_{2}}^{- 1} + C$

Step 12.2.3

Multiply $\frac{1}{8}$ by $1$ .

$- \frac{1}{8 u_{1}} + \frac{1}{8} {u_{2}}^{- 1} + C$

Step 12.2.4

Combine $\frac{1}{8}$ and ${u_{2}}^{- 1}$ .

$- \frac{1}{8 u_{1}} + \frac{{u_{2}}^{- 1}}{8} + C$

Step 12.2.5

Move ${u_{2}}^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{8 u_{1}} + \frac{1}{8 u_{2}} + C$

Step 13

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 13.1

Replace all occurrences of $u_{1}$ with $x - 1$ .

$- \frac{1}{8 (x - 1)} + \frac{1}{8 u_{2}} + C$

Step 13.2

Replace all occurrences of $u_{2}$ with $x + 3$ .

$- \frac{1}{8 (x - 1)} + \frac{1}{8 (x + 3)} + C$