Calculus Examples

$\int x \sin^{3} (x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sin^{3} (x)$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

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

Step 5

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

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

Step 6

Factor out $\cos^{2} (x)$ .

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

Step 7

Using the Pythagorean Identity, rewrite $\cos^{2} (x)$ as $1 - \sin^{2} (x)$ .

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

Step 8

Let $u = \sin (x)$ . Then $d u = \cos (x) d x$ , so $\frac{1}{\cos (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = \sin (x)$ . Find $\frac{d u}{d x}$ .

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Step 8.1.1

Differentiate $\sin (x)$ .

$\frac{d}{d x} [\sin (x)]$

Step 8.1.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\cos (x)$

Step 8.2

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

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

Step 9

Split the single integral into multiple integrals.

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

Step 10

Apply the constant rule.

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

Step 11

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

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

Step 12

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

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (- (\sin (x) + C) + \frac{1}{3} (u + C - (\frac{1}{3} u^{3} + C)))$

Step 13

Simplify.

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (- \sin (x) + \frac{1}{3} (u - \frac{1}{3} u^{3})) + C$

Step 14

Replace all occurrences of $u$ with $\sin (x)$ .

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (- \sin (x) + \frac{1}{3} (\sin (x) - \frac{1}{3} \sin^{3} (x))) + C$

Step 15

Simplify.

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Step 15.1

Simplify each term.

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Step 15.1.1

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

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (- \sin (x) + \frac{1}{3} (\sin (x) - \frac{\sin^{3} (x)}{3})) + C$

Step 15.1.2

Apply the distributive property.

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (- \sin (x) + \frac{1}{3} \sin (x) + \frac{1}{3} (- \frac{\sin^{3} (x)}{3})) + C$

Step 15.1.3

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

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (- \sin (x) + \frac{\sin (x)}{3} + \frac{1}{3} (- \frac{\sin^{3} (x)}{3})) + C$

Step 15.1.4

Multiply $\frac{1}{3} (- \frac{\sin^{3} (x)}{3})$ .

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Step 15.1.4.1

Multiply $\frac{1}{3}$ by $\frac{\sin^{3} (x)}{3}$ .

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (- \sin (x) + \frac{\sin (x)}{3} - \frac{\sin^{3} (x)}{3 \cdot 3}) + C$

Step 15.1.4.2

Multiply $3$ by $3$ .

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (- \sin (x) + \frac{\sin (x)}{3} - \frac{\sin^{3} (x)}{9}) + C$

Step 15.2

To write $- \sin (x)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (- \sin (x) \cdot \frac{3}{3} + \frac{\sin (x)}{3} - \frac{\sin^{3} (x)}{9}) + C$

Step 15.3

Combine $- \sin (x)$ and $\frac{3}{3}$ .

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (\frac{- \sin (x) \cdot 3}{3} + \frac{\sin (x)}{3} - \frac{\sin^{3} (x)}{9}) + C$

Step 15.4

Combine the numerators over the common denominator.

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (\frac{- \sin (x) \cdot 3 + \sin (x)}{3} - \frac{\sin^{3} (x)}{9}) + C$

Step 15.5

Simplify each term.

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Step 15.5.1

Simplify the numerator.

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Step 15.5.1.1

Factor $\sin (x)$ out of $- \sin (x) \cdot 3 + \sin (x)$ .

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Step 15.5.1.1.1

Factor $\sin (x)$ out of $- \sin (x) \cdot 3$ .

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (\frac{\sin (x) (- 1 \cdot 3) + \sin (x)}{3} - \frac{\sin^{3} (x)}{9}) + C$

Step 15.5.1.1.2

Multiply by $1$ .

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (\frac{\sin (x) (- 1 \cdot 3) + \sin (x) \cdot 1}{3} - \frac{\sin^{3} (x)}{9}) + C$

Step 15.5.1.1.3

Factor $\sin (x)$ out of $\sin (x) (- 1 \cdot 3) + \sin (x) \cdot 1$ .

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (\frac{\sin (x) (- 1 \cdot 3 + 1)}{3} - \frac{\sin^{3} (x)}{9}) + C$

Step 15.5.1.2

Multiply $- 1$ by $3$ .

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (\frac{\sin (x) (- 3 + 1)}{3} - \frac{\sin^{3} (x)}{9}) + C$

Step 15.5.1.3

Add $- 3$ and $1$ .

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (\frac{\sin (x) \cdot - 2}{3} - \frac{\sin^{3} (x)}{9}) + C$

Step 15.5.2

Move $- 2$ to the left of $\sin (x)$ .

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (\frac{- 2 \cdot \sin (x)}{3} - \frac{\sin^{3} (x)}{9}) + C$

Step 15.5.3

Move the negative in front of the fraction.

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - (- \frac{2 \sin (x)}{3} - \frac{\sin^{3} (x)}{9}) + C$

Step 15.6

Apply the distributive property.

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) - - \frac{2 \sin (x)}{3} - - \frac{\sin^{3} (x)}{9} + C$

Step 15.7

Multiply $- - \frac{2 \sin (x)}{3}$ .

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Step 15.7.1

Multiply $- 1$ by $- 1$ .

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) + 1 \frac{2 \sin (x)}{3} - - \frac{\sin^{3} (x)}{9} + C$

Step 15.7.2

Multiply $\frac{2 \sin (x)}{3}$ by $1$ .

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) + \frac{2 \sin (x)}{3} - - \frac{\sin^{3} (x)}{9} + C$

Step 15.8

Multiply $- - \frac{\sin^{3} (x)}{9}$ .

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Step 15.8.1

Multiply $- 1$ by $- 1$ .

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) + \frac{2 \sin (x)}{3} + 1 \frac{\sin^{3} (x)}{9} + C$

Step 15.8.2

Multiply $\frac{\sin^{3} (x)}{9}$ by $1$ .

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) + \frac{2 \sin (x)}{3} + \frac{\sin^{3} (x)}{9} + C$

Step 16

Reorder terms.

$x (- \cos (x) + \frac{1}{3} \cos^{3} (x)) + \frac{2}{3} \sin (x) + \frac{1}{9} \sin^{3} (x) + C$