Calculus Examples

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

Step 1

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

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

Step 2

Apply the distributive property.

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

Step 3

Apply the distributive property.

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

Step 4

Apply the distributive property.

$\int \frac{x^{3}}{x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1} d x$

Step 5

Reorder $x$ and $1$ .

$\int \frac{x^{3}}{x \cdot x + 1 \cdot x + 1 x + 1 \cdot 1} d x$

Step 6

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

$\int \frac{x^{3}}{x^{1} x + 1 x + 1 x + 1 \cdot 1} d x$

Step 7

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

$\int \frac{x^{3}}{x^{1} x^{1} + 1 x + 1 x + 1 \cdot 1} d x$

Step 8

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

$\int \frac{x^{3}}{x^{1 + 1} + 1 x + 1 x + 1 \cdot 1} d x$

Step 9

Simplify the expression.

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Step 9.1

Add $1$ and $1$ .

$\int \frac{x^{3}}{x^{2} + 1 x + 1 x + 1 \cdot 1} d x$

Step 9.2

Multiply $x$ by $1$ .

$\int \frac{x^{3}}{x^{2} + x + 1 x + 1 \cdot 1} d x$

Step 9.3

Multiply $x$ by $1$ .

$\int \frac{x^{3}}{x^{2} + x + x + 1 \cdot 1} d x$

Step 9.4

Multiply $1$ by $1$ .

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

Step 10

Add $x$ and $x$ .

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

Step 11

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

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Step 11.1

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

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 11.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x^{2}$ .

$x$

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 11.3

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$2 x^{2}$

$x$

Step 11.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 2 x^{2} + x$

$x$

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$2 x^{2}$

$x$

Step 11.5

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

$x$

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$x$

Step 11.6

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

$x$

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$x$

$0$

Step 11.7

Divide the highest order term in the dividend $- 2 x^{2}$ by the highest order term in divisor $x^{2}$ .

$x$

$2$

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$x$

$0$

Step 11.8

Multiply the new quotient term by the divisor.

						$x$	-	$2$
$x^{2}$	+	$2 x$	+	$1$		$x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$0$
					-	$x^{3}$	-	$2 x^{2}$	-	$x$
							-	$2 x^{2}$	-	$x$	+	$0$
							-	$2 x^{2}$	-	$4 x$	-	$2$

Step 11.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{2} - 4 x - 2$

$x$

$2$

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$x$

$0$

$2 x^{2}$

$4 x$

$2$

Step 11.10

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

$x$

$2$

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$x$

$0$

$2 x^{2}$

$4 x$

$2$

$3 x$

$2$

Step 11.11

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

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

Step 12

Split the single integral into multiple integrals.

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

Step 13

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

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

Step 14

Apply the constant rule.

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

Step 15

Write the fraction using partial fraction decomposition.

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Step 15.1

Decompose the fraction and multiply through by the common denominator.

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Step 15.1.1

Factor using the perfect square rule.

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Step 15.1.1.1

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

$\frac{3 x + 2}{x^{2} + 2 x + 1^{2}}$

Step 15.1.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 15.1.1.3

Rewrite the polynomial.

$\frac{3 x + 2}{x^{2} + 2 \cdot x \cdot 1 + 1^{2}}$

Step 15.1.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 15.1.2

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 15.1.3

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

Step 15.1.4

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

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

Step 15.1.5

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

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Step 15.1.5.1

Cancel the common factor.

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

Step 15.1.5.2

Divide $3 x + 2$ by $1$ .

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

Step 15.1.6

Simplify each term.

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Step 15.1.6.1

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

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Step 15.1.6.1.1

Cancel the common factor.

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

Step 15.1.6.1.2

Divide $A$ by $1$ .

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

Step 15.1.6.2

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

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Step 15.1.6.2.1

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

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

Step 15.1.6.2.2

Cancel the common factors.

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Step 15.1.6.2.2.1

Multiply by $1$ .

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

Step 15.1.6.2.2.2

Cancel the common factor.

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

Step 15.1.6.2.2.3

Rewrite the expression.

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

Step 15.1.6.2.2.4

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

$3 x + 2 = A + B (x + 1)$

Step 15.1.6.3

Apply the distributive property.

$3 x + 2 = A + B x + B \cdot 1$

Step 15.1.6.4

Multiply $B$ by $1$ .

$3 x + 2 = A + B x + B$

Step 15.1.7

Reorder $A$ and $B x$ .

$3 x + 2 = B x + A + B$

Step 15.2

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

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Step 15.2.1

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.

$3 = B$

Step 15.2.2

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.

$2 = A + B$

Step 15.2.3

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

$3 = B$

$2 = A + B$

$3 = B$

$2 = A + B$

Step 15.3

Solve the system of equations.

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Step 15.3.1

Rewrite the equation as $B = 3$ .

$B = 3$

$2 = A + B$

Step 15.3.2

Replace all occurrences of $B$ with $3$ in each equation.

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Step 15.3.2.1

Replace all occurrences of $B$ in $2 = A + B$ with $3$ .

$2 = A + 3$

$B = 3$

Step 15.3.2.2

Simplify the left side.

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Step 15.3.2.2.1

Remove parentheses.

$2 = A + 3$

$B = 3$

$2 = A + 3$

$B = 3$

$2 = A + 3$

$B = 3$

Step 15.3.3

Solve for $A$ in $2 = A + 3$ .

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Step 15.3.3.1

Rewrite the equation as $A + 3 = 2$ .

$A + 3 = 2$

$B = 3$

Step 15.3.3.2

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

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Step 15.3.3.2.1

Subtract $3$ from both sides of the equation.

$A = 2 - 3$

$B = 3$

Step 15.3.3.2.2

Subtract $3$ from $2$ .

$A = - 1$

$B = 3$

$A = - 1$

$B = 3$

$A = - 1$

$B = 3$

Step 15.3.4

Solve the system of equations.

$A = - 1 B = 3$

Step 15.3.5

List all of the solutions.

$A = - 1, B = 3$

Step 15.4

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

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

Step 15.5

Move the negative in front of the fraction.

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

Step 16

Split the single integral into multiple integrals.

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

Step 17

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

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

Step 18

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

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Step 18.1

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

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Step 18.1.1

Differentiate $x + 1$ .

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

Step 18.1.2

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 18.1.3

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 18.1.4

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

$1 + 0$

Step 18.1.5

Add $1$ and $0$ .

$1$

Step 18.2

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

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

Step 19

Apply basic rules of exponents.

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Step 19.1

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

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

Step 19.2

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

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Step 19.2.1

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

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

Step 19.2.2

Multiply $2$ by $- 1$ .

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

Step 20

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

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

Step 21

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 22

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

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Step 22.1

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

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Step 22.1.1

Differentiate $x + 1$ .

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

Step 22.1.2

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 22.1.3

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 22.1.4

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

$1 + 0$

Step 22.1.5

Add $1$ and $0$ .

$1$

Step 22.2

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

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

Step 23

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

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

Step 24

Simplify.

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

Step 25

Substitute back in for each integration substitution variable.

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Step 25.1

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

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

Step 25.2

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

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

Step 26

Reorder terms.

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