Calculus Examples

$\int \frac{5 x^{5} + 5 x^{2} + 10}{x^{3} - x} d x$

Step 1

Divide $5 x^{5} + 5 x^{2} + 10$ by $x^{3} - x$ .

Tap for more steps...

Step 1.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x^{3}$

$0 x^{2}$

$x$

$0$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$5 x^{2}$

$0 x$

$10$

Step 1.2

Divide the highest order term in the dividend $5 x^{5}$ by the highest order term in divisor $x^{3}$ .

$5 x^{2}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$5 x^{2}$

$0 x$

$10$

Step 1.3

Multiply the new quotient term by the divisor.

$5 x^{2}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$5 x^{2}$

$0 x$

$10$

$5 x^{5}$

$0$

$5 x^{3}$

$0$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{5} + 0 - 5 x^{3} + 0$

$5 x^{2}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$5 x^{2}$

$0 x$

$10$

$5 x^{5}$

$0$

$5 x^{3}$

$0$

Step 1.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$5 x^{2}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$5 x^{2}$

$0 x$

$10$

$5 x^{5}$

$0$

$5 x^{3}$

$0$

$5 x^{3}$

$5 x^{2}$

Step 1.6

Pull the next term from the original dividend down into the current dividend.

$5 x^{2}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$5 x^{2}$

$0 x$

$10$

$5 x^{5}$

$0$

$5 x^{3}$

$0$

$5 x^{3}$

$5 x^{2}$

$0 x$

$10$

Step 1.7

Divide the highest order term in the dividend $5 x^{3}$ by the highest order term in divisor $x^{3}$ .

$5 x^{2}$

$0 x$

$5$

$x^{3}$

$0 x^{2}$

$x$

$0$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$5 x^{2}$

$0 x$

$10$

$5 x^{5}$

$0$

$5 x^{3}$

$0$

$5 x^{3}$

$5 x^{2}$

$0 x$

$10$

Step 1.8

Multiply the new quotient term by the divisor.

$5 x^{2}$

$0 x$

$5$

$x^{3}$

$0 x^{2}$

$x$

$0$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$5 x^{2}$

$0 x$

$10$

$5 x^{5}$

$0$

$5 x^{3}$

$0$

$5 x^{3}$

$5 x^{2}$

$0 x$

$10$

$5 x^{3}$

$0$

$5 x$

$0$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{3} + 0 - 5 x + 0$

$5 x^{2}$

$0 x$

$5$

$x^{3}$

$0 x^{2}$

$x$

$0$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$5 x^{2}$

$0 x$

$10$

$5 x^{5}$

$0$

$5 x^{3}$

$0$

$5 x^{3}$

$5 x^{2}$

$0 x$

$10$

$5 x^{3}$

$0$

$5 x$

$0$

Step 1.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$5 x^{2}$

$0 x$

$5$

$x^{3}$

$0 x^{2}$

$x$

$0$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$5 x^{2}$

$0 x$

$10$

$5 x^{5}$

$0$

$5 x^{3}$

$0$

$5 x^{3}$

$5 x^{2}$

$0 x$

$10$

$5 x^{3}$

$0$

$5 x$

$0$

$5 x^{2}$

$5 x$

$10$

Step 1.11

The final answer is the quotient plus the remainder over the divisor.

$\int 5 x^{2} + 5 + \frac{5 x^{2} + 5 x + 10}{x^{3} - x} d x$

Step 2

Split the single integral into multiple integrals.

$\int 5 x^{2} d x + \int 5 d x + \int \frac{5 x^{2} + 5 x + 10}{x^{3} - x} d x$

Step 3

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int x^{2} d x + \int 5 d x + \int \frac{5 x^{2} + 5 x + 10}{x^{3} - x} d x$

Step 4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$5 (\frac{1}{3} x^{3} + C) + \int 5 d x + \int \frac{5 x^{2} + 5 x + 10}{x^{3} - x} d x$

Step 5

Apply the constant rule.

$5 (\frac{1}{3} x^{3} + C) + 5 x + C + \int \frac{5 x^{2} + 5 x + 10}{x^{3} - x} d x$

Step 6

Combine $\frac{1}{3}$ and $x^{3}$ .

$5 (\frac{x^{3}}{3} + C) + 5 x + C + \int \frac{5 x^{2} + 5 x + 10}{x^{3} - x} d x$

Step 7

Write the fraction using partial fraction decomposition.

Tap for more steps...

Step 7.1

Decompose the fraction and multiply through by the common denominator.

Tap for more steps...

Step 7.1.1

Factor the fraction.

Tap for more steps...

Step 7.1.1.1

Factor $5$ out of $5 x^{2} + 5 x + 10$ .

Tap for more steps...

Step 7.1.1.1.1

Factor $5$ out of $5 x^{2}$ .

$\frac{5 (x^{2}) + 5 x + 10}{x^{3} - x}$

Step 7.1.1.1.2

Factor $5$ out of $5 x$ .

$\frac{5 (x^{2}) + 5 (x) + 10}{x^{3} - x}$

Step 7.1.1.1.3

Factor $5$ out of $10$ .

$\frac{5 (x^{2}) + 5 x + 5 \cdot 2}{x^{3} - x}$

Step 7.1.1.1.4

Factor $5$ out of $5 (x^{2}) + 5 x$ .

$\frac{5 (x^{2} + x) + 5 \cdot 2}{x^{3} - x}$

Step 7.1.1.1.5

Factor $5$ out of $5 (x^{2} + x) + 5 \cdot 2$ .

$\frac{5 (x^{2} + x + 2)}{x^{3} - x}$

Step 7.1.1.2

Factor $x$ out of $x^{3} - x$ .

Tap for more steps...

Step 7.1.1.2.1

Factor $x$ out of $x^{3}$ .

$\frac{5 (x^{2} + x + 2)}{x \cdot x^{2} - x}$

Step 7.1.1.2.2

Factor $x$ out of $- x$ .

$\frac{5 (x^{2} + x + 2)}{x \cdot x^{2} + x \cdot - 1}$

Step 7.1.1.2.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 1$ .

$\frac{5 (x^{2} + x + 2)}{x (x^{2} - 1)}$

Step 7.1.1.3

Rewrite $1$ as $1^{2}$ .

$\frac{5 (x^{2} + x + 2)}{x (x^{2} - 1^{2})}$

Step 7.1.1.4

Factor.

Tap for more steps...

Step 7.1.1.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{5 (x^{2} + x + 2)}{x ((x + 1) (x - 1))}$

Step 7.1.1.4.2

Remove unnecessary parentheses.

$\frac{5 (x^{2} + x + 2)}{x (x + 1) (x - 1)}$

Step 7.1.2

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 $B$ .

$\frac{A}{x} + \frac{B}{x + 1}$

Step 7.1.3

$\frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x - 1}$

Step 7.1.4

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

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

Step 7.1.5

Cancel the common factor of $x$ .

Tap for more steps...

Step 7.1.5.1

Cancel the common factor.

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

Step 7.1.5.2

Rewrite the expression.

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

Step 7.1.6

Cancel the common factor of $x + 1$ .

Tap for more steps...

Step 7.1.6.1

Cancel the common factor.

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

Step 7.1.6.2

Rewrite the expression.

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

Step 7.1.7

Cancel the common factor of $x - 1$ .

Tap for more steps...

Step 7.1.7.1

Cancel the common factor.

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

Step 7.1.7.2

Divide $5 (x^{2} + x + 2)$ by $1$ .

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

Step 7.1.8

Apply the distributive property.

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

Step 7.1.9

Multiply $5$ by $2$ .

$5 x^{2} + 5 x + 10 = \frac{A (x (x + 1) (x - 1))}{x} + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10

Simplify each term.

Tap for more steps...

Step 7.1.10.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 7.1.10.1.1

Cancel the common factor.

$5 x^{2} + 5 x + 10 = \frac{A (x (x + 1) (x - 1))}{x} + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.1.2

Divide $A ((x + 1) (x - 1))$ by $1$ .

$5 x^{2} + 5 x + 10 = A ((x + 1) (x - 1)) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.2

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

Tap for more steps...

Step 7.1.10.2.1

Apply the distributive property.

$5 x^{2} + 5 x + 10 = A (x (x - 1) + 1 (x - 1)) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.2.2

Apply the distributive property.

$5 x^{2} + 5 x + 10 = A (x \cdot x + x \cdot - 1 + 1 (x - 1)) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.2.3

Apply the distributive property.

$5 x^{2} + 5 x + 10 = A (x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.3

Simplify and combine like terms.

Tap for more steps...

Step 7.1.10.3.1

Simplify each term.

Tap for more steps...

Step 7.1.10.3.1.1

Multiply $x$ by $x$ .

$5 x^{2} + 5 x + 10 = A (x^{2} + x \cdot - 1 + 1 x + 1 \cdot - 1) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.3.1.2

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

$5 x^{2} + 5 x + 10 = A (x^{2} - 1 \cdot x + 1 x + 1 \cdot - 1) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.3.1.3

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

$5 x^{2} + 5 x + 10 = A (x^{2} - x + 1 x + 1 \cdot - 1) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.3.1.4

Multiply $x$ by $1$ .

$5 x^{2} + 5 x + 10 = A (x^{2} - x + x + 1 \cdot - 1) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.3.1.5

Multiply $- 1$ by $1$ .

$5 x^{2} + 5 x + 10 = A (x^{2} - x + x - 1) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.3.2

Add $- x$ and $x$ .

$5 x^{2} + 5 x + 10 = A (x^{2} + 0 - 1) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.3.3

Add $x^{2}$ and $0$ .

$5 x^{2} + 5 x + 10 = A (x^{2} - 1) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.4

Apply the distributive property.

$5 x^{2} + 5 x + 10 = A x^{2} + A \cdot - 1 + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.5

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

$5 x^{2} + 5 x + 10 = A x^{2} - 1 \cdot A + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.6

Rewrite $- 1 A$ as $- A$ .

$5 x^{2} + 5 x + 10 = A x^{2} - A + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.7

Cancel the common factor of $x + 1$ .

Tap for more steps...

Step 7.1.10.7.1

Cancel the common factor.

$5 x^{2} + 5 x + 10 = A x^{2} - A + \frac{B (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.7.2

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

$5 x^{2} + 5 x + 10 = A x^{2} - A + (B) (x (x - 1)) + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.8

Apply the distributive property.

$5 x^{2} + 5 x + 10 = A x^{2} - A + B (x \cdot x + x \cdot - 1) + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.9

Multiply $x$ by $x$ .

$5 x^{2} + 5 x + 10 = A x^{2} - A + B (x^{2} + x \cdot - 1) + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.10

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

$5 x^{2} + 5 x + 10 = A x^{2} - A + B (x^{2} - 1 \cdot x) + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.11

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

$5 x^{2} + 5 x + 10 = A x^{2} - A + B (x^{2} - x) + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.12

Apply the distributive property.

$5 x^{2} + 5 x + 10 = A x^{2} - A + B x^{2} + B (- x) + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.13

Rewrite using the commutative property of multiplication.

$5 x^{2} + 5 x + 10 = A x^{2} - A + B x^{2} - B x + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.14

Cancel the common factor of $x - 1$ .

Tap for more steps...

Step 7.1.10.14.1

Cancel the common factor.

$5 x^{2} + 5 x + 10 = A x^{2} - A + B x^{2} - B x + \frac{C (x (x + 1) (x - 1))}{x - 1}$

Step 7.1.10.14.2

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

$5 x^{2} + 5 x + 10 = A x^{2} - A + B x^{2} - B x + (C) (x (x + 1))$

Step 7.1.10.15

Apply the distributive property.

$5 x^{2} + 5 x + 10 = A x^{2} - A + B x^{2} - B x + C (x \cdot x + x \cdot 1)$

Step 7.1.10.16

Multiply $x$ by $x$ .

$5 x^{2} + 5 x + 10 = A x^{2} - A + B x^{2} - B x + C (x^{2} + x \cdot 1)$

Step 7.1.10.17

Multiply $x$ by $1$ .

$5 x^{2} + 5 x + 10 = A x^{2} - A + B x^{2} - B x + C (x^{2} + x)$

Step 7.1.10.18

Apply the distributive property.

$5 x^{2} + 5 x + 10 = A x^{2} - A + B x^{2} - B x + C x^{2} + C x$

Step 7.1.11

Simplify the expression.

Tap for more steps...

Step 7.1.11.1

Move $B$ .

$5 x^{2} + 5 x + 10 = A x^{2} - A + B x^{2} - x B + C x^{2} + C x$

Step 7.1.11.2

Move $- A$ .

$5 x^{2} + 5 x + 10 = A x^{2} + B x^{2} - x B + C x^{2} + C x - A$

Step 7.1.11.3

Move $- x B$ .

$5 x^{2} + 5 x + 10 = A x^{2} + B x^{2} + C x^{2} - x B + C x - A$

Step 7.2

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

Tap for more steps...

Step 7.2.1

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.

$5 = A + B + C$

Step 7.2.2

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.

$5 = - 1 B + C$

Step 7.2.3

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.

$10 = - 1 A$

Step 7.2.4

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

$5 = A + B + C$

$5 = - 1 B + C$

$10 = - 1 A$

$5 = A + B + C$

$5 = - 1 B + C$

$10 = - 1 A$

Step 7.3

Solve the system of equations.

Tap for more steps...

Step 7.3.1

Solve for $A$ in $10 = - 1 A$ .

Tap for more steps...

Step 7.3.1.1

Rewrite the equation as $- 1 A = 10$ .

$- 1 A = 10$

$5 = A + B + C$

$5 = - 1 B + C$

Step 7.3.1.2

Divide each term in $- 1 A = 10$ by $- 1$ and simplify.

Tap for more steps...

Step 7.3.1.2.1

Divide each term in $- 1 A = 10$ by $- 1$ .

$\frac{- 1 A}{- 1} = \frac{10}{- 1}$

$5 = A + B + C$

$5 = - 1 B + C$

Step 7.3.1.2.2

Simplify the left side.

Tap for more steps...

Step 7.3.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{A}{1} = \frac{10}{- 1}$

$5 = A + B + C$

$5 = - 1 B + C$

Step 7.3.1.2.2.2

Divide $A$ by $1$ .

$A = \frac{10}{- 1}$

$5 = A + B + C$

$5 = - 1 B + C$

$A = \frac{10}{- 1}$

$5 = A + B + C$

$5 = - 1 B + C$

Step 7.3.1.2.3

Simplify the right side.

Tap for more steps...

Step 7.3.1.2.3.1

Divide $10$ by $- 1$ .

$A = - 10$

$5 = A + B + C$

$5 = - 1 B + C$

$A = - 10$

$5 = A + B + C$

$5 = - 1 B + C$

$A = - 10$

$5 = A + B + C$

$5 = - 1 B + C$

$A = - 10$

$5 = A + B + C$

$5 = - 1 B + C$

Step 7.3.2

Replace all occurrences of $A$ with $- 10$ in each equation.

Tap for more steps...

Step 7.3.2.1

Replace all occurrences of $A$ in $5 = A + B + C$ with $- 10$ .

$5 = (- 10) + B + C$

$A = - 10$

$5 = - 1 B + C$

Step 7.3.2.2

Simplify the right side.

Tap for more steps...

Step 7.3.2.2.1

Remove parentheses.

$5 = - 10 + B + C$

$A = - 10$

$5 = - 1 B + C$

$5 = - 10 + B + C$

$A = - 10$

$5 = - 1 B + C$

$5 = - 10 + B + C$

$A = - 10$

$5 = - 1 B + C$

Step 7.3.3

Solve for $B$ in $5 = - 10 + B + C$ .

Tap for more steps...

Step 7.3.3.1

Rewrite the equation as $- 10 + B + C = 5$ .

$- 10 + B + C = 5$

$A = - 10$

$5 = - 1 B + C$

Step 7.3.3.2

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

Tap for more steps...

Step 7.3.3.2.1

Add $10$ to both sides of the equation.

$B + C = 5 + 10$

$A = - 10$

$5 = - 1 B + C$

Step 7.3.3.2.2

Subtract $C$ from both sides of the equation.

$B = 5 + 10 - C$

$A = - 10$

$5 = - 1 B + C$

Step 7.3.3.2.3

Add $5$ and $10$ .

$B = 15 - C$

$A = - 10$

$5 = - 1 B + C$

$B = 15 - C$

$A = - 10$

$5 = - 1 B + C$

$B = 15 - C$

$A = - 10$

$5 = - 1 B + C$

Step 7.3.4

Replace all occurrences of $B$ with $15 - C$ in each equation.

Tap for more steps...

Step 7.3.4.1

Replace all occurrences of $B$ in $5 = - 1 B + C$ with $15 - C$ .

$5 = - 1 (15 - C) + C$

$B = 15 - C$

$A = - 10$

Step 7.3.4.2

Simplify the right side.

Tap for more steps...

Step 7.3.4.2.1

Simplify $- 1 (15 - C) + C$ .

Tap for more steps...

Step 7.3.4.2.1.1

Simplify each term.

Tap for more steps...

Step 7.3.4.2.1.1.1

Apply the distributive property.

$5 = - 1 \cdot 15 - 1 (- C) + C$

$B = 15 - C$

$A = - 10$

Step 7.3.4.2.1.1.2

Multiply $- 1$ by $15$ .

$5 = - 15 - 1 (- C) + C$

$B = 15 - C$

$A = - 10$

Step 7.3.4.2.1.1.3

Multiply $- 1 (- C)$ .

Tap for more steps...

Step 7.3.4.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$5 = - 15 + 1 C + C$

$B = 15 - C$

$A = - 10$

Step 7.3.4.2.1.1.3.2

Multiply $C$ by $1$ .

$5 = - 15 + C + C$

$B = 15 - C$

$A = - 10$

$5 = - 15 + C + C$

$B = 15 - C$

$A = - 10$

$5 = - 15 + C + C$

$B = 15 - C$

$A = - 10$

Step 7.3.4.2.1.2

Add $C$ and $C$ .

$5 = - 15 + 2 C$

$B = 15 - C$

$A = - 10$

$5 = - 15 + 2 C$

$B = 15 - C$

$A = - 10$

$5 = - 15 + 2 C$

$B = 15 - C$

$A = - 10$

$5 = - 15 + 2 C$

$B = 15 - C$

$A = - 10$

Step 7.3.5

Solve for $C$ in $5 = - 15 + 2 C$ .

Tap for more steps...

Step 7.3.5.1

Rewrite the equation as $- 15 + 2 C = 5$ .

$- 15 + 2 C = 5$

$B = 15 - C$

$A = - 10$

Step 7.3.5.2

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

Tap for more steps...

Step 7.3.5.2.1

Add $15$ to both sides of the equation.

$2 C = 5 + 15$

$B = 15 - C$

$A = - 10$

Step 7.3.5.2.2

Add $5$ and $15$ .

$2 C = 20$

$B = 15 - C$

$A = - 10$

$2 C = 20$

$B = 15 - C$

$A = - 10$

Step 7.3.5.3

Divide each term in $2 C = 20$ by $2$ and simplify.

Tap for more steps...

Step 7.3.5.3.1

Divide each term in $2 C = 20$ by $2$ .

$\frac{2 C}{2} = \frac{20}{2}$

$B = 15 - C$

$A = - 10$

Step 7.3.5.3.2

Simplify the left side.

Tap for more steps...

Step 7.3.5.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.3.5.3.2.1.1

Cancel the common factor.

$\frac{2 C}{2} = \frac{20}{2}$

$B = 15 - C$

$A = - 10$

Step 7.3.5.3.2.1.2

Divide $C$ by $1$ .

$C = \frac{20}{2}$

$B = 15 - C$

$A = - 10$

$C = \frac{20}{2}$

$B = 15 - C$

$A = - 10$

$C = \frac{20}{2}$

$B = 15 - C$

$A = - 10$

Step 7.3.5.3.3

Simplify the right side.

Tap for more steps...

Step 7.3.5.3.3.1

Divide $20$ by $2$ .

$C = 10$

$B = 15 - C$

$A = - 10$

$C = 10$

$B = 15 - C$

$A = - 10$

$C = 10$

$B = 15 - C$

$A = - 10$

$C = 10$

$B = 15 - C$

$A = - 10$

Step 7.3.6

Replace all occurrences of $C$ with $10$ in each equation.

Tap for more steps...

Step 7.3.6.1

Replace all occurrences of $C$ in $B = 15 - C$ with $10$ .

$B = 15 - (10)$

$C = 10$

$A = - 10$

Step 7.3.6.2

Simplify the right side.

Tap for more steps...

Step 7.3.6.2.1

Simplify $15 - (10)$ .

Tap for more steps...

Step 7.3.6.2.1.1

Multiply $- 1$ by $10$ .

$B = 15 - 10$

$C = 10$

$A = - 10$

Step 7.3.6.2.1.2

Subtract $10$ from $15$ .

$B = 5$

$C = 10$

$A = - 10$

$B = 5$

$C = 10$

$A = - 10$

$B = 5$

$C = 10$

$A = - 10$

$B = 5$

$C = 10$

$A = - 10$

Step 7.3.7

List all of the solutions.

$B = 5, C = 10, A = - 10$

Step 7.4

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

$- \frac{10}{x} + \frac{5}{x + 1} + \frac{10}{x - 1}$

Step 7.5

Move the negative in front of the fraction.

$5 (\frac{x^{3}}{3} + C) + 5 x + C + \int - \frac{10}{x} + \frac{5}{x + 1} + \frac{10}{x - 1} d x$

Step 8

Split the single integral into multiple integrals.

$5 (\frac{x^{3}}{3} + C) + 5 x + C + \int - \frac{10}{x} d x + \int \frac{5}{x + 1} d x + \int \frac{10}{x - 1} d x$

Step 9

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

$5 (\frac{x^{3}}{3} + C) + 5 x + C - \int \frac{10}{x} d x + \int \frac{5}{x + 1} d x + \int \frac{10}{x - 1} d x$

Step 10

Since $10$ is constant with respect to $x$ , move $10$ out of the integral.

$5 (\frac{x^{3}}{3} + C) + 5 x + C - (10 \int \frac{1}{x} d x) + \int \frac{5}{x + 1} d x + \int \frac{10}{x - 1} d x$

Step 11

Multiply $10$ by $- 1$ .

$5 (\frac{x^{3}}{3} + C) + 5 x + C - 10 \int \frac{1}{x} d x + \int \frac{5}{x + 1} d x + \int \frac{10}{x - 1} d x$

Step 12

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$5 (\frac{x^{3}}{3} + C) + 5 x + C - 10 (\ln (| x |) + C) + \int \frac{5}{x + 1} d x + \int \frac{10}{x - 1} d x$

Step 13

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 (\frac{x^{3}}{3} + C) + 5 x + C - 10 (\ln (| x |) + C) + 5 \int \frac{1}{x + 1} d x + \int \frac{10}{x - 1} d x$

Step 14

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

Tap for more steps...

Step 14.1

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

Tap for more steps...

Step 14.1.1

Differentiate $x + 1$ .

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

Step 14.1.2

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 14.1.3

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 14.1.4

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

$1 + 0$

Step 14.1.5

Add $1$ and $0$ .

$1$

Step 14.2

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

$5 (\frac{x^{3}}{3} + C) + 5 x + C - 10 (\ln (| x |) + C) + 5 \int \frac{1}{u_{1}} d u_{1} + \int \frac{10}{x - 1} d x$

Step 15

The integral of $\frac{1}{u_{1}}$ with respect to $u_{1}$ is $\ln (| u_{1} |)$ .

$5 (\frac{x^{3}}{3} + C) + 5 x + C - 10 (\ln (| x |) + C) + 5 (\ln (| u_{1} |) + C) + \int \frac{10}{x - 1} d x$

Step 16

Since $10$ is constant with respect to $x$ , move $10$ out of the integral.

$5 (\frac{x^{3}}{3} + C) + 5 x + C - 10 (\ln (| x |) + C) + 5 (\ln (| u_{1} |) + C) + 10 \int \frac{1}{x - 1} d x$

Step 17

Let $u_{2} = x - 1$ . Then $d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 17.1

Let $u_{2} = x - 1$ . Find $\frac{d u_{2}}{d x}$ .

Tap for more steps...

Step 17.1.1

Differentiate $x - 1$ .

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

Step 17.1.2

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 17.1.3

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 17.1.4

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

$1 + 0$

Step 17.1.5

Add $1$ and $0$ .

$1$

Step 17.2

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

$5 (\frac{x^{3}}{3} + C) + 5 x + C - 10 (\ln (| x |) + C) + 5 (\ln (| u_{1} |) + C) + 10 \int \frac{1}{u_{2}} d u_{2}$

Step 18

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$5 (\frac{x^{3}}{3} + C) + 5 x + C - 10 (\ln (| x |) + C) + 5 (\ln (| u_{1} |) + C) + 10 (\ln (| u_{2} |) + C)$

Step 19

Simplify.

$\frac{5 x^{3}}{3} + 5 x - 10 \ln (| x |) + 5 \ln (| u_{1} |) + 10 \ln (| u_{2} |) + C$

Step 20

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 20.1

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

$\frac{5 x^{3}}{3} + 5 x - 10 \ln (| x |) + 5 \ln (| x + 1 |) + 10 \ln (| u_{2} |) + C$

Step 20.2

Replace all occurrences of $u_{2}$ with $x - 1$ .

$\frac{5 x^{3}}{3} + 5 x - 10 \ln (| x |) + 5 \ln (| x + 1 |) + 10 \ln (| x - 1 |) + C$

Step 21

Reorder terms.

$\frac{5}{3} x^{3} + 5 x - 10 \ln (| x |) + 5 \ln (| x + 1 |) + 10 \ln (| x - 1 |) + C$