Calculus Examples

$\int x^{2} \arctan (x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \arctan (x)$ and $d v = x^{2}$ .

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

Combine $\arctan (x)$ and $\frac{x^{3}}{3}$ .

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

Step 3

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

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

Step 4

Combine $x^{3}$ and $\frac{1}{x^{2} + 1}$ .

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

Step 5

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

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Step 5.1

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

$x^{2}$

$0 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 5.2

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

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 5.3

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$x$

Step 5.4

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

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$x$

Step 5.5

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

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$x$

Step 5.6

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

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$x$

$0$

Step 5.7

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

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

Move the negative in front of the fraction.

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

Step 8

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

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

Step 9

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

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

Step 10

Let $u = x^{2} + 1$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 10.1

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

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Step 10.1.1

Differentiate $x^{2} + 1$ .

$\frac{d}{d x} [x^{2} + 1]$

Step 10.1.2

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$

Step 10.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [1]$

Step 10.1.4

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

$2 x + 0$

Step 10.1.5

Add $2 x$ and $0$ .

$2 x$

Step 10.2

Rewrite the problem using $u$ and $d u$ .

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

Step 11

Simplify.

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Step 11.1

Multiply $\frac{1}{u}$ by $\frac{1}{2}$ .

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

Step 11.2

Move $2$ to the left of $u$ .

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

Step 12

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 13

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

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

Step 14

Simplify.

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Step 14.1

Simplify.

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

Step 14.2

Simplify.

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Step 14.2.1

Combine $\frac{1}{3}$ and $\arctan (x)$ .

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

Step 14.2.2

Combine $\frac{\arctan (x)}{3}$ and $x^{3}$ .

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

Step 14.2.3

To write $- \frac{1}{3} (\frac{1}{2} x^{2} - \frac{1}{2} \ln (| u |))$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 14.2.4

Combine $- \frac{1}{3} (\frac{1}{2} x^{2} - \frac{1}{2} \ln (| u |))$ and $\frac{3}{3}$ .

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

Step 14.2.5

Combine the numerators over the common denominator.

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

Step 14.2.6

Combine $\frac{1}{2}$ and $x^{2}$ .

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

Step 14.2.7

Combine $\ln (| u |)$ and $\frac{1}{2}$ .

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

Step 14.2.8

Multiply $3$ by $- 1$ .

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

Step 14.2.9

Combine $- 3$ and $\frac{1}{3}$ .

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

Step 14.2.10

Cancel the common factor of $- 3$ and $3$ .

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Step 14.2.10.1

Factor $3$ out of $- 3$ .

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

Step 14.2.10.2

Cancel the common factors.

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Step 14.2.10.2.1

Factor $3$ out of $3$ .

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

Step 14.2.10.2.2

Cancel the common factor.

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

Step 14.2.10.2.3

Rewrite the expression.

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

Step 14.2.10.2.4

Divide $- 1$ by $1$ .

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

Step 15

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

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

Step 16

Combine the numerators over the common denominator.

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

Step 17

Reorder terms.

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