Calculus Examples

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

Step 1

Write $\cos^{3} (2 x)$ as a function.

$f (x) = \cos^{3} (2 x)$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

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

Step 4

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

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Differentiate $2 x$ .

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

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

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

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

$2 \cdot 1$

Multiply $2$ by $1$ .

$2$

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

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

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

$\cos (u_{1})$

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

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

Step 10

Split the single integral into multiple integrals.

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

Step 11

Apply the constant rule.

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

Step 12

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

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

Step 13

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

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

Step 14

Simplify.

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

Step 15

Substitute back in for each integration substitution variable.

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

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

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

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

Step 16

Simplify.

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

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

Apply the distributive property.

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

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

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

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

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

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

Multiply $2$ by $3$ .

$\frac{\sin (2 x)}{2} - \frac{\sin^{3} (2 x)}{6} + C$

Step 17

Reorder terms.

$\frac{1}{2} \sin (2 x) - \frac{1}{6} \sin^{3} (2 x) + C$

Step 18

The answer is the antiderivative of the function $f (x) = \cos^{3} (2 x)$ .

$F (x) =$ $\frac{1}{2} \sin (2 x) - \frac{1}{6} \sin^{3} (2 x) + C$