Calculus Examples

$\cos^{3} (2 x) \sin^{2} (2 x) d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $2 x$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

$2 \cdot 1$

Step 1.1.4

Multiply $2$ by $1$ .

$2$

Step 1.2

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

$\int \cos^{3} (u_{1}) \sin^{2} (u_{1}) \frac{1}{2} d u_{1}$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{2}$ and $\cos^{3} (u_{1})$ .

$\int \frac{\cos^{3} (u_{1})}{2} \sin^{2} (u_{1}) d u_{1}$

Step 2.2

Combine $\frac{\cos^{3} (u_{1})}{2}$ and $\sin^{2} (u_{1})$ .

$\int \frac{\cos^{3} (u_{1}) \sin^{2} (u_{1})}{2} d u_{1}$

Step 3

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

$\frac{1}{2} \int \cos^{3} (u_{1}) \sin^{2} (u_{1}) d u_{1}$

Step 4

Factor out $\cos^{2} (u_{1})$ .

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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Step 6.1.1

Differentiate $\sin (u_{1})$ .

$\frac{d}{d u_{1}} [\sin (u_{1})]$

Step 6.1.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$\cos (u_{1})$

Step 6.2

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

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

Step 7

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

$\frac{1}{2} \int 1 {u_{2}}^{2} - {u_{2}}^{2} {u_{2}}^{2} d u_{2}$

Step 8

Simplify.

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Step 8.1

Multiply ${u_{2}}^{2}$ by $1$ .

$\frac{1}{2} \int {u_{2}}^{2} - {u_{2}}^{2} {u_{2}}^{2} d u_{2}$

Step 8.2

Multiply ${u_{2}}^{2}$ by ${u_{2}}^{2}$ by adding the exponents.

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Step 8.2.1

Move ${u_{2}}^{2}$ .

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

Step 8.2.2

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

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

Step 8.2.3

Add $2$ and $2$ .

$\frac{1}{2} \int {u_{2}}^{2} - {u_{2}}^{4} d u_{2}$

Step 9

Split the single integral into multiple integrals.

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

Step 10

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

$\frac{1}{2} (\frac{1}{3} {u_{2}}^{3} + C + \int - {u_{2}}^{4} d u_{2})$

Step 11

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

$\frac{1}{2} (\frac{1}{3} {u_{2}}^{3} + C - \int {u_{2}}^{4} d u_{2})$

Step 12

By the Power Rule, the integral of ${u_{2}}^{4}$ with respect to $u_{2}$ is $\frac{1}{5} {u_{2}}^{5}$ .

$\frac{1}{2} (\frac{1}{3} {u_{2}}^{3} + C - (\frac{1}{5} {u_{2}}^{5} + C))$

Step 13

Simplify.

$\frac{1}{2} (\frac{1}{3} {u_{2}}^{3} - \frac{1}{5} {u_{2}}^{5}) + C$

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

Replace all occurrences of $u_{2}$ with $\sin (u_{1})$ .

$\frac{1}{2} (\frac{1}{3} \sin^{3} (u_{1}) - \frac{1}{5} \sin^{5} (u_{1})) + C$

Step 14.2

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

$\frac{1}{2} (\frac{1}{3} \sin^{3} (2 x) - \frac{1}{5} \sin^{5} (2 x)) + C$