Calculus Examples

$6 \cos^{3} (2 x)$

Step 1

Write $6 \cos^{3} (2 x)$ as a function.

$f (x) = 6 \cos^{3} (2 x)$

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.

$F (x) = \int 6 \cos^{3} (2 x) d x$

Step 4

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int \cos^{3} (2 x) d x$

Step 5

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 5.1

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

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Step 5.1.1

Differentiate $2 x$ .

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

Step 5.1.2

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 5.1.3

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 5.1.4

Multiply $2$ by $1$ .

$2$

Step 5.2

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

$6 \int \cos^{3} (u_{1}) \frac{1}{2} d u_{1}$

Step 6

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

$6 \int \frac{\cos^{3} (u_{1})}{2} d u_{1}$

Step 7

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

$6 (\frac{1}{2} \int \cos^{3} (u_{1}) d u_{1})$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{2}$ and $6$ .

$\frac{6}{2} \int \cos^{3} (u_{1}) d u_{1}$

Step 8.2

Cancel the common factor of $6$ and $2$ .

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Step 8.2.1

Factor $2$ out of $6$ .

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

Step 8.2.2

Cancel the common factors.

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Step 8.2.2.1

Factor $2$ out of $2$ .

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

Step 8.2.2.2

Cancel the common factor.

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

Step 8.2.2.3

Rewrite the expression.

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

Step 8.2.2.4

Divide $3$ by $1$ .

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

Step 9

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

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

Step 10

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

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

Step 11

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 11.1

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

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Step 11.1.1

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

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

Step 11.1.2

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

$\cos (u_{1})$

Step 11.2

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

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

Step 12

Split the single integral into multiple integrals.

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

Step 13

Apply the constant rule.

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

Step 14

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

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

Step 15

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

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

Step 16

Simplify.

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

Step 17

Substitute back in for each integration substitution variable.

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Step 17.1

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

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

Step 17.2

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

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

Step 18

Simplify.

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Step 18.1

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

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

Step 18.2

Apply the distributive property.

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

Step 18.3

Cancel the common factor of $3$ .

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Step 18.3.1

Move the leading negative in $- \frac{\sin^{3} (2 x)}{3}$ into the numerator.

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

Step 18.3.2

Cancel the common factor.

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

Step 18.3.3

Rewrite the expression.

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

Step 19

The answer is the antiderivative of the function $f (x) = 6 \cos^{3} (2 x)$ .

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