Calculus Examples

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

Step 1

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

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Step 1.1

Let $u_{2} = \cos (2 x)$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $\cos (2 x)$ .

$\frac{d}{d x} [\cos (2 x)]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 2 x$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [2 x]$

Step 1.1.2.2

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

$- \sin (u_{1}) \frac{d}{d x} [2 x]$

Step 1.1.2.3

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

$- \sin (2 x) \frac{d}{d x} [2 x]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

$- \sin (2 x) (2 \frac{d}{d x} [x])$

Step 1.1.3.2

Multiply $2$ by $- 1$ .

$- 2 \sin (2 x) \frac{d}{d x} [x]$

Step 1.1.3.3

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

$- 2 \sin (2 x) \cdot 1$

Step 1.1.3.4

Multiply $- 2$ by $1$ .

$- 2 \sin (2 x)$

Step 1.2

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

$\int {u_{2}}^{3} \frac{1}{- 2} d u_{2}$

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

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

Step 2.2

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

$\int - \frac{{u_{2}}^{3}}{2} d u_{2}$

Step 3

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

$- \int \frac{{u_{2}}^{3}}{2} d u_{2}$

Step 4

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

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

Step 5

By the Power Rule, the integral of ${u_{2}}^{3}$ with respect to $u_{2}$ is $\frac{1}{4} {u_{2}}^{4}$ .

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

Step 6

Simplify.

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Step 6.1

Rewrite $- \frac{1}{2} (\frac{1}{4} {u_{2}}^{4} + C)$ as $- \frac{1}{2} \cdot \frac{1}{4} {u_{2}}^{4} + C$ .

$- \frac{1}{2} \cdot \frac{1}{4} {u_{2}}^{4} + C$

Step 6.2

Simplify.

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Step 6.2.1

Multiply $\frac{1}{4}$ by $\frac{1}{2}$ .

$- \frac{1}{4 \cdot 2} {u_{2}}^{4} + C$

Step 6.2.2

Multiply $4$ by $2$ .

$- \frac{1}{8} {u_{2}}^{4} + C$

Step 7

Replace all occurrences of $u_{2}$ with $\cos (2 x)$ .

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