Calculus Examples

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

Step 1

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

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Step 1.1

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

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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 $u$ and $d u$ .

$\int 2 u \sin (u) \frac{1}{\frac{1}{2}} d u$

Step 2

Simplify.

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Step 2.1

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

$\int 2 u \sin (u) (1 \cdot 2) d u$

Step 2.2

Multiply $2$ by $1$ .

$\int 2 u \sin (u) \cdot 2 d u$

Step 2.3

Multiply $2$ by $2$ .

$\int 4 u \sin (u) d u$

Step 3

Since $4$ is constant with respect to $u$ , move $4$ out of the integral.

$4 \int u \sin (u) d u$

Step 4

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = u$ and $dv = \sin (u)$ .

$4 (u (- \cos (u)) - \int - \cos (u) d u)$

Step 5

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$4 (u (- \cos (u)) - - \int \cos (u) d u)$

Step 6

Simplify.

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Step 6.1

Multiply $- 1$ by $- 1$ .

$4 (u (- \cos (u)) + 1 \int \cos (u) d u)$

Step 6.2

Multiply $\int \cos (u) d u$ by $1$ .

$4 (u (- \cos (u)) + \int \cos (u) d u)$

Step 7

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$4 (u (- \cos (u)) + \sin (u) + C)$

Step 8

Rewrite $4 (u (- \cos (u)) + \sin (u) + C)$ as $4 (- u \cos (u) + \sin (u)) + C$ .

$4 (- u \cos (u) + \sin (u)) + C$

Step 9

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

$4 (- \frac{x}{2} \cos (\frac{x}{2}) + \sin (\frac{x}{2})) + C$

Step 10

Simplify.

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Step 10.1

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

$4 (- \frac{\cos (\frac{x}{2}) x}{2} + \sin (\frac{x}{2})) + C$

Step 10.2

Apply the distributive property.

$4 (- \frac{\cos (\frac{x}{2}) x}{2}) + 4 \sin (\frac{x}{2}) + C$

Step 10.3

Cancel the common factor of $2$ .

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Step 10.3.1

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

$4 \frac{- \cos (\frac{x}{2}) x}{2} + 4 \sin (\frac{x}{2}) + C$

Step 10.3.2

Factor $2$ out of $4$ .

$2 (2) \frac{- \cos (\frac{x}{2}) x}{2} + 4 \sin (\frac{x}{2}) + C$

Step 10.3.3

Cancel the common factor.

$2 \cdot 2 \frac{- \cos (\frac{x}{2}) x}{2} + 4 \sin (\frac{x}{2}) + C$

Step 10.3.4

Rewrite the expression.

$2 (- \cos (\frac{x}{2}) x) + 4 \sin (\frac{x}{2}) + C$

Step 10.4

Multiply $- 1$ by $2$ .

$- 2 (\cos (\frac{x}{2}) x) + 4 \sin (\frac{x}{2}) + C$

Step 10.5

Reorder factors in $- 2 \cos (\frac{x}{2}) x + 4 \sin (\frac{x}{2})$ .

$- 2 x \cos (\frac{x}{2}) + 4 \sin (\frac{x}{2}) + C$

Step 11

Reorder terms.

$- 2 x \cos (\frac{1}{2} x) + 4 \sin (\frac{1}{2} x) + C$