Calculus Examples

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

Step 1

Decompose the fraction and multiply through by the common denominator.

Tap for more steps...

Step 1.1

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 $A$ .

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

Step 1.2

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

Step 1.3

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

Step 1.4

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

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

Step 1.5

Cancel the common factor of ${(x + 1)}^{3}$ .

Tap for more steps...

Step 1.5.1

Cancel the common factor.

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

Step 1.5.2

Divide $2 x - 1$ by $1$ .

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

Step 1.6

Simplify each term.

Tap for more steps...

Step 1.6.1

Cancel the common factor of ${(x + 1)}^{3}$ .

Tap for more steps...

Step 1.6.1.1

Cancel the common factor.

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

Step 1.6.1.2

Divide $A$ by $1$ .

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

Step 1.6.2

Cancel the common factor of ${(x + 1)}^{3}$ and ${(x + 1)}^{2}$ .

Tap for more steps...

Step 1.6.2.1

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

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

Step 1.6.2.2

Cancel the common factors.

Tap for more steps...

Step 1.6.2.2.1

Multiply by $1$ .

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

Step 1.6.2.2.2

Cancel the common factor.

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

Step 1.6.2.2.3

Rewrite the expression.

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

Step 1.6.2.2.4

Divide $B (x + 1)$ by $1$ .

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

Step 1.6.3

Apply the distributive property.

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

Step 1.6.4

Multiply $B$ by $1$ .

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

Step 1.6.5

Cancel the common factor of ${(x + 1)}^{3}$ and $x + 1$ .

Tap for more steps...

Step 1.6.5.1

Factor $x + 1$ out of $(C) {(x + 1)}^{3}$ .

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

Step 1.6.5.2

Cancel the common factors.

Tap for more steps...

Step 1.6.5.2.1

Multiply by $1$ .

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

Step 1.6.5.2.2

Cancel the common factor.

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

Step 1.6.5.2.3

Rewrite the expression.

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

Step 1.6.5.2.4

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

$2 x - 1 = A + B x + B + C {(x + 1)}^{2}$

Step 1.6.6

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

$2 x - 1 = A + B x + B + C ((x + 1) (x + 1))$

Step 1.6.7

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

Tap for more steps...

Step 1.6.7.1

Apply the distributive property.

$2 x - 1 = A + B x + B + C (x (x + 1) + 1 (x + 1))$

Step 1.6.7.2

Apply the distributive property.

$2 x - 1 = A + B x + B + C (x \cdot x + x \cdot 1 + 1 (x + 1))$

Step 1.6.7.3

Apply the distributive property.

$2 x - 1 = A + B x + B + C (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1)$

Step 1.6.8

Simplify and combine like terms.

Tap for more steps...

Step 1.6.8.1

Simplify each term.

Tap for more steps...

Step 1.6.8.1.1

Multiply $x$ by $x$ .

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

Step 1.6.8.1.2

Multiply $x$ by $1$ .

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

Step 1.6.8.1.3

Multiply $x$ by $1$ .

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

Step 1.6.8.1.4

Multiply $1$ by $1$ .

$2 x - 1 = A + B x + B + C (x^{2} + x + x + 1)$

Step 1.6.8.2

Add $x$ and $x$ .

$2 x - 1 = A + B x + B + C (x^{2} + 2 x + 1)$

Step 1.6.9

Apply the distributive property.

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

Step 1.6.10

Simplify.

Tap for more steps...

Step 1.6.10.1

Rewrite using the commutative property of multiplication.

$2 x - 1 = A + B x + B + C x^{2} + 2 C x + C \cdot 1$

Step 1.6.10.2

Multiply $C$ by $1$ .

$2 x - 1 = A + B x + B + C x^{2} + 2 C x + C$

Step 1.7

Simplify the expression.

Tap for more steps...

Step 1.7.1

Move $B$ .

$2 x - 1 = A + B x + C x^{2} + 2 C x + B + C$

Step 1.7.2

Move $A$ .

$2 x - 1 = B x + C x^{2} + 2 C x + A + B + C$

Step 1.7.3

Reorder $B x$ and $C x^{2}$ .

$2 x - 1 = C x^{2} + B x + 2 C x + A + B + C$

Step 2

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

Tap for more steps...

Step 2.1

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.

$0 = C$

Step 2.2

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.

$2 = B + 2 C$

Step 2.3

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.

$- 1 = A + B + C$

Step 2.4

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

$0 = C$

$2 = B + 2 C$

$- 1 = A + B + C$

$0 = C$

$2 = B + 2 C$

$- 1 = A + B + C$

Step 3

Solve the system of equations.

Tap for more steps...

Step 3.1

Rewrite the equation as $C = 0$ .

$C = 0$

$2 = B + 2 C$

$- 1 = A + B + C$

Step 3.2

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

Tap for more steps...

Step 3.2.1

Replace all occurrences of $C$ in $2 = B + 2 C$ with $0$ .

$2 = B + 2 (0)$

$C = 0$

$- 1 = A + B + C$

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Simplify $B + 2 (0)$ .

Tap for more steps...

Step 3.2.2.1.1

Multiply $2$ by $0$ .

$2 = B + 0$

$C = 0$

$- 1 = A + B + C$

Step 3.2.2.1.2

Add $B$ and $0$ .

$2 = B$

$C = 0$

$- 1 = A + B + C$

$2 = B$

$C = 0$

$- 1 = A + B + C$

$2 = B$

$C = 0$

$- 1 = A + B + C$

Step 3.2.3

Replace all occurrences of $C$ in $- 1 = A + B + C$ with $0$ .

$- 1 = A + B + 0$

$2 = B$

$C = 0$

Step 3.2.4

Simplify $- 1 = A + B + 0$ .

Tap for more steps...

Step 3.2.4.1

Simplify the left side.

Tap for more steps...

Step 3.2.4.1.1

Remove parentheses.

$- 1 = A + B + 0$

$2 = B$

$C = 0$

$- 1 = A + B + 0$

$2 = B$

$C = 0$

Step 3.2.4.2

Simplify the right side.

Tap for more steps...

Step 3.2.4.2.1

Add $A + B$ and $0$ .

$- 1 = A + B$

$2 = B$

$C = 0$

$- 1 = A + B$

$2 = B$

$C = 0$

$- 1 = A + B$

$2 = B$

$C = 0$

$- 1 = A + B$

$2 = B$

$C = 0$

Step 3.3

Rewrite the equation as $B = 2$ .

$B = 2$

$- 1 = A + B$

$C = 0$

Step 3.4

Replace all occurrences of $B$ with $2$ in each equation.

Tap for more steps...

Step 3.4.1

Replace all occurrences of $B$ in $- 1 = A + B$ with $2$ .

$- 1 = A + 2$

$B = 2$

$C = 0$

Step 3.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.1

Remove parentheses.

$- 1 = A + 2$

$B = 2$

$C = 0$

$- 1 = A + 2$

$B = 2$

$C = 0$

$- 1 = A + 2$

$B = 2$

$C = 0$

Step 3.5

Solve for $A$ in $- 1 = A + 2$ .

Tap for more steps...

Step 3.5.1

Rewrite the equation as $A + 2 = - 1$ .

$A + 2 = - 1$

$B = 2$

$C = 0$

Step 3.5.2

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

Tap for more steps...

Step 3.5.2.1

Subtract $2$ from both sides of the equation.

$A = - 1 - 2$

$B = 2$

$C = 0$

Step 3.5.2.2

Subtract $2$ from $- 1$ .

$A = - 3$

$B = 2$

$C = 0$

$A = - 3$

$B = 2$

$C = 0$

$A = - 3$

$B = 2$

$C = 0$

Step 3.6

Solve the system of equations.

$A = - 3 B = 2 C = 0$

Step 3.7

List all of the solutions.

$A = - 3, B = 2, C = 0$

Step 4

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

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