Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Differentiate $\frac{x}{2}$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

$\frac{1}{2} \cdot 1$

Step 1.1.4

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

$\frac{1}{2}$

Step 1.2

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

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

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

Step 2.2

Multiply $2$ by $1$ .

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

Step 2.3

Move $2$ to the left of $(1 - \cos (u_{1})) \sin (u_{1})$ .

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

Step 3

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

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

Step 4

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

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

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

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

Step 4.1.2

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

$- \sin (u_{1})$

Step 4.2

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

$2 \int - 1 + u_{2} d u_{2}$

Step 5

Split the single integral into multiple integrals.

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

Step 6

Apply the constant rule.

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

Step 7

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

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

Step 8

Simplify.

Tap for more steps...

Step 8.1

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

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

Step 8.2

Simplify.

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

Step 9

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 9.1

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

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

Step 9.2

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

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

Step 10

Simplify.

Tap for more steps...

Step 10.1

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

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

Step 10.2

Apply the distributive property.

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

Step 10.3

Multiply $- 1$ by $2$ .

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

Step 10.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.4.1

Cancel the common factor.

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

Step 10.4.2

Rewrite the expression.

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

Step 11

Reorder terms.

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