Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $\sin (\frac{x}{2})$ .

$\frac{d}{d x} [\sin (\frac{x}{2})]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = \frac{x}{2}$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $\frac{x}{2}$ .

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [\frac{x}{2}]$

Step 1.1.2.2

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

$\cos (u_{1}) \frac{d}{d x} [\frac{x}{2}]$

Step 1.1.2.3

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

$\cos (\frac{x}{2}) \frac{d}{d x} [\frac{x}{2}]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

$\cos (\frac{x}{2}) (\frac{1}{2} \frac{d}{d x} [x])$

Step 1.1.3.2

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

$\frac{\cos (\frac{x}{2})}{2} \frac{d}{d x} [x]$

Step 1.1.3.3

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

$\frac{\cos (\frac{x}{2})}{2} \cdot 1$

Step 1.1.3.4

Multiply $\frac{\cos (\frac{x}{2})}{2}$ by $1$ .

$\frac{\cos (\frac{x}{2})}{2}$

Step 1.2

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

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

Step 2

Divide $2$ by $1$ .

$\int 2 u_{2} d u_{2}$

Step 3

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

$2 \int u_{2} d u_{2}$

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Rewrite $2 (\frac{1}{2} {u_{2}}^{2} + C)$ as $2 (\frac{1}{2}) {u_{2}}^{2} + C$ .

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

Step 5.2

Simplify.

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Step 5.2.1

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

$\frac{2}{2} {u_{2}}^{2} + C$

Step 5.2.2

Cancel the common factor of $2$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{2}{2} {u_{2}}^{2} + C$

Step 5.2.2.2

Rewrite the expression.

$1 {u_{2}}^{2} + C$

Step 5.2.3

Multiply ${u_{2}}^{2}$ by $1$ .

${u_{2}}^{2} + C$

Step 6

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

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

Step 7

Reorder terms.

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