Calculus Examples

$\int \cos^{3} (4 x) d x$

Step 1

Let $u_{1} = 4 x$ . Then $d u_{1} = 4 d x$ , so $\frac{1}{4} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 1.1

Let $u_{1} = 4 x$ . Find $\frac{d u_{1}}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 1.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

Step 1.1.3

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

$4 \cdot 1$

Step 1.1.4

Multiply $4$ by $1$ .

$4$

Step 1.2

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

$\int \cos^{3} (u_{1}) \frac{1}{4} d u_{1}$

Step 2

Combine $\cos^{3} (u_{1})$ and $\frac{1}{4}$ .

$\int \frac{\cos^{3} (u_{1})}{4} d u_{1}$

Step 3

Since $\frac{1}{4}$ is constant with respect to $u_{1}$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int \cos^{3} (u_{1}) d u_{1}$

Step 4

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

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

Step 5

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

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

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 $d$ $u_{2}$ .

Tap for more steps...

Step 6.1

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

Tap for more steps...

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 $d u_{2}$ .

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

Step 7

Split the single integral into multiple integrals.

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

Step 8

Apply the constant rule.

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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

Step 12

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 12.1

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

$\frac{1}{4} (\sin (u_{1}) - \frac{1}{3} \sin^{3} (u_{1})) + C$

Step 12.2

Replace all occurrences of $u_{1}$ with $4 x$ .

$\frac{1}{4} (\sin (4 x) - \frac{1}{3} \sin^{3} (4 x)) + C$

Step 13

Simplify.

Tap for more steps...

Step 13.1

Combine $\sin^{3} (4 x)$ and $\frac{1}{3}$ .

$\frac{1}{4} (\sin (4 x) - \frac{\sin^{3} (4 x)}{3}) + C$

Step 13.2

Apply the distributive property.

$\frac{1}{4} \sin (4 x) + \frac{1}{4} (- \frac{\sin^{3} (4 x)}{3}) + C$

Step 13.3

Combine $\frac{1}{4}$ and $\sin (4 x)$ .

$\frac{\sin (4 x)}{4} + \frac{1}{4} (- \frac{\sin^{3} (4 x)}{3}) + C$

Step 13.4

Multiply $\frac{1}{4} (- \frac{\sin^{3} (4 x)}{3})$ .

Tap for more steps...

Step 13.4.1

Multiply $\frac{1}{4}$ by $\frac{\sin^{3} (4 x)}{3}$ .

$\frac{\sin (4 x)}{4} - \frac{\sin^{3} (4 x)}{4 \cdot 3} + C$

Step 13.4.2

Multiply $4$ by $3$ .

$\frac{\sin (4 x)}{4} - \frac{\sin^{3} (4 x)}{12} + C$

Step 14

Reorder terms.

$\frac{1}{4} \sin (4 x) - \frac{1}{12} \sin^{3} (4 x) + C$