Calculus Examples

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

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.

Tap for more steps...

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 $x^{2} + 2 x + 1$ .

Tap for more steps...

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.

Tap for more steps...

Step 15.1

Decompose the fraction and multiply through by the common denominator.

Tap for more steps...

Step 15.1.1

Factor using the perfect square rule.

Tap for more steps...

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

Tap for more steps...

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.

Tap for more steps...

Step 15.1.6.1

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

Tap for more steps...

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 $x + 1$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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 $2 = A + 3$ .

Tap for more steps...

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.

Tap for more steps...

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 $d$ $u_{1}$ .

Tap for more steps...

Step 18.1

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

Tap for more steps...

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.

Tap for more steps...

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 ${({u_{1}}^{2})}^{- 1}$ .

Tap for more steps...

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 $d$ $u_{2}$ .

Tap for more steps...

Step 22.1

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

Tap for more steps...

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.

Tap for more steps...

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.

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