Calculus Examples

$\int 3 \cos^{2} (5 x) d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int \cos^{2} (5 x) d x$

Step 2

Let $u_{1} = 5 x$ . Then $d u_{1} = 5 d x$ , so $\frac{1}{5} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 2.1

Let $u_{1} = 5 x$ . Find $\frac{d u_{1}}{d x}$ .

Tap for more steps...

Step 2.1.1

Differentiate $5 x$ .

$\frac{d}{d x} [5 x]$

Step 2.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x]$

Step 2.1.3

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

$5 \cdot 1$

Step 2.1.4

Multiply $5$ by $1$ .

$5$

Step 2.2

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

$3 \int \cos^{2} (u_{1}) \frac{1}{5} d u_{1}$

Step 3

Combine $\cos^{2} (u_{1})$ and $\frac{1}{5}$ .

$3 \int \frac{\cos^{2} (u_{1})}{5} d u_{1}$

Step 4

Since $\frac{1}{5}$ is constant with respect to $u_{1}$ , move $\frac{1}{5}$ out of the integral.

$3 (\frac{1}{5} \int \cos^{2} (u_{1}) d u_{1})$

Step 5

Combine $\frac{1}{5}$ and $3$ .

$\frac{3}{5} \int \cos^{2} (u_{1}) d u_{1}$

Step 6

Use the half-angle formula to rewrite $\cos^{2} (u_{1})$ as $\frac{1 + \cos (2 u_{1})}{2}$ .

$\frac{3}{5} \int \frac{1 + \cos (2 u_{1})}{2} d u_{1}$

Step 7

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

$\frac{3}{5} (\frac{1}{2} \int 1 + \cos (2 u_{1}) d u_{1})$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Multiply $\frac{1}{2}$ by $\frac{3}{5}$ .

$\frac{3}{2 \cdot 5} \int 1 + \cos (2 u_{1}) d u_{1}$

Step 8.2

Multiply $2$ by $5$ .

$\frac{3}{10} \int 1 + \cos (2 u_{1}) d u_{1}$

Step 9

Split the single integral into multiple integrals.

$\frac{3}{10} (\int d u_{1} + \int \cos (2 u_{1}) d u_{1})$

Step 10

Apply the constant rule.

$\frac{3}{10} (u_{1} + C + \int \cos (2 u_{1}) d u_{1})$

Step 11

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

Tap for more steps...

Step 11.1

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

Tap for more steps...

Step 11.1.1

Differentiate $2 u_{1}$ .

$\frac{d}{d u_{1}} [2 u_{1}]$

Step 11.1.2

Since $2$ is constant with respect to $u_{1}$ , the derivative of $2 u_{1}$ with respect to $u_{1}$ is $2 \frac{d}{d u_{1}} [u_{1}]$ .

$2 \frac{d}{d u_{1}} [u_{1}]$

Step 11.1.3

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

$2 \cdot 1$

Step 11.1.4

Multiply $2$ by $1$ .

$2$

Step 11.2

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

$\frac{3}{10} (u_{1} + C + \int \cos (u_{2}) \frac{1}{2} d u_{2})$

Step 12

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

$\frac{3}{10} (u_{1} + C + \int \frac{\cos (u_{2})}{2} d u_{2})$

Step 13

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

$\frac{3}{10} (u_{1} + C + \frac{1}{2} \int \cos (u_{2}) d u_{2})$

Step 14

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$\frac{3}{10} (u_{1} + C + \frac{1}{2} (\sin (u_{2}) + C))$

Step 15

Simplify.

$\frac{3}{10} (u_{1} + \frac{1}{2} \sin (u_{2})) + C$

Step 16

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 16.1

Replace all occurrences of $u_{1}$ with $5 x$ .

$\frac{3}{10} (5 x + \frac{1}{2} \sin (u_{2})) + C$

Step 16.2

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

$\frac{3}{10} (5 x + \frac{1}{2} \sin (2 u_{1})) + C$

Step 16.3

Replace all occurrences of $u_{1}$ with $5 x$ .

$\frac{3}{10} (5 x + \frac{1}{2} \sin (2 (5 x))) + C$

Step 17

Simplify.

Tap for more steps...

Step 17.1

Simplify each term.

Tap for more steps...

Step 17.1.1

Multiply $5$ by $2$ .

$\frac{3}{10} (5 x + \frac{1}{2} \sin (10 x)) + C$

Step 17.1.2

Combine $\frac{1}{2}$ and $\sin (10 x)$ .

$\frac{3}{10} (5 x + \frac{\sin (10 x)}{2}) + C$

Step 17.2

Apply the distributive property.

$\frac{3}{10} (5 x) + \frac{3}{10} \cdot \frac{\sin (10 x)}{2} + C$

Step 17.3

Cancel the common factor of $5$ .

Tap for more steps...

Step 17.3.1

Factor $5$ out of $10$ .

$\frac{3}{5 (2)} (5 x) + \frac{3}{10} \cdot \frac{\sin (10 x)}{2} + C$

Step 17.3.2

Factor $5$ out of $5 x$ .

$\frac{3}{5 (2)} (5 (x)) + \frac{3}{10} \cdot \frac{\sin (10 x)}{2} + C$

Step 17.3.3

Cancel the common factor.

$\frac{3}{5 \cdot 2} (5 x) + \frac{3}{10} \cdot \frac{\sin (10 x)}{2} + C$

Step 17.3.4

Rewrite the expression.

$\frac{3}{2} x + \frac{3}{10} \cdot \frac{\sin (10 x)}{2} + C$

Step 17.4

Combine $\frac{3}{2}$ and $x$ .

$\frac{3 x}{2} + \frac{3}{10} \cdot \frac{\sin (10 x)}{2} + C$

Step 17.5

Multiply $\frac{3}{10} \cdot \frac{\sin (10 x)}{2}$ .

Tap for more steps...

Step 17.5.1

Multiply $\frac{3}{10}$ by $\frac{\sin (10 x)}{2}$ .

$\frac{3 x}{2} + \frac{3 \sin (10 x)}{10 \cdot 2} + C$

Step 17.5.2

Multiply $10$ by $2$ .

$\frac{3 x}{2} + \frac{3 \sin (10 x)}{20} + C$

Step 18

Reorder terms.

$\frac{3}{2} x + \frac{3}{20} \sin (10 x) + C$