Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $x^{2}$ .

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

Step 1.1.2

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

$2 x$

Step 1.2

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

$\int \sin^{2} (u_{1}) \frac{1}{2} d u_{1}$

Step 2

Combine $\sin^{2} (u_{1})$ and $\frac{1}{2}$ .

$\int \frac{\sin^{2} (u_{1})}{2} d u_{1}$

Step 3

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

$\frac{1}{2} \int \sin^{2} (u_{1}) d u_{1}$

Step 4

Use the half-angle formula to rewrite $\sin^{2} (u_{1})$ as $\frac{1 - \cos (2 u_{1})}{2}$ .

$\frac{1}{2} \int \frac{1 - \cos (2 u_{1})}{2} d u_{1}$

Step 5

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

$\frac{1}{2} (\frac{1}{2} \int 1 - \cos (2 u_{1}) d u_{1})$

Step 6

Simplify.

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Step 6.1

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

$\frac{1}{2 \cdot 2} \int 1 - \cos (2 u_{1}) d u_{1}$

Step 6.2

Multiply $2$ by $2$ .

$\frac{1}{4} \int 1 - \cos (2 u_{1}) d u_{1}$

Step 7

Split the single integral into multiple integrals.

$\frac{1}{4} (\int d u_{1} + \int - \cos (2 u_{1}) d u_{1})$

Step 8

Apply the constant rule.

$\frac{1}{4} (u_{1} + C + \int - \cos (2 u_{1}) d u_{1})$

Step 9

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

$\frac{1}{4} (u_{1} + C - \int \cos (2 u_{1}) d u_{1})$

Step 10

Let $u_{2} = 2 u_{1}$ . Then $d u_{2} = 2 d u_{1}$ , so $\frac{1}{2} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 10.1

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

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Step 10.1.1

Differentiate $2 u_{1}$ .

$\frac{d}{d u_{1}} [2 u_{1}]$

Step 10.1.2

Since $2$ is constant with respect to $u_{1}$ , the derivative of $2 u_{1}$ with respect to $u_{1}$ is $2 \frac{d}{d u_{1}} [u_{1}]$ .

$2 \frac{d}{d u_{1}} [u_{1}]$

Step 10.1.3

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

$2 \cdot 1$

Step 10.1.4

Multiply $2$ by $1$ .

$2$

Step 10.2

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

$\frac{1}{4} (u_{1} + C - \int \cos (u_{2}) \frac{1}{2} d u_{2})$

Step 11

Combine $\cos (u_{2})$ and $\frac{1}{2}$ .

$\frac{1}{4} (u_{1} + C - \int \frac{\cos (u_{2})}{2} d u_{2})$

Step 12

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

$\frac{1}{4} (u_{1} + C - (\frac{1}{2} \int \cos (u_{2}) d u_{2}))$

Step 13

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$\frac{1}{4} (u_{1} + C - \frac{1}{2} (\sin (u_{2}) + C))$

Step 14

Simplify.

$\frac{1}{4} (u_{1} - \frac{1}{2} \sin (u_{2})) + C$

Step 15

Substitute back in for each integration substitution variable.

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Step 15.1

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

$\frac{1}{4} (x^{2} - \frac{1}{2} \sin (u_{2})) + C$

Step 15.2

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

$\frac{1}{4} (x^{2} - \frac{1}{2} \sin (2 u_{1})) + C$

Step 15.3

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

$\frac{1}{4} (x^{2} - \frac{1}{2} \sin (2 x^{2})) + C$

Step 16

Simplify.

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Step 16.1

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

$\frac{1}{4} (x^{2} - \frac{\sin (2 x^{2})}{2}) + C$

Step 16.2

Apply the distributive property.

$\frac{1}{4} x^{2} + \frac{1}{4} (- \frac{\sin (2 x^{2})}{2}) + C$

Step 16.3

Combine $\frac{1}{4}$ and $x^{2}$ .

$\frac{x^{2}}{4} + \frac{1}{4} (- \frac{\sin (2 x^{2})}{2}) + C$

Step 16.4

Multiply $\frac{1}{4} (- \frac{\sin (2 x^{2})}{2})$ .

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Step 16.4.1

Multiply $\frac{1}{4}$ by $\frac{\sin (2 x^{2})}{2}$ .

$\frac{x^{2}}{4} - \frac{\sin (2 x^{2})}{4 \cdot 2} + C$

Step 16.4.2

Multiply $4$ by $2$ .

$\frac{x^{2}}{4} - \frac{\sin (2 x^{2})}{8} + C$

Step 17

Reorder terms.

$\frac{1}{4} x^{2} - \frac{1}{8} \sin (2 x^{2}) + C$