Calculus Examples

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

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = \cos (\frac{1}{2} x)$ .

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $2$ by $2$ .

$x^{2} (2 \sin (\frac{x}{2})) - \int 4 \sin (\frac{x}{2}) x d x$

Step 3

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

$x^{2} (2 \sin (\frac{x}{2})) - (4 \int \sin (\frac{x}{2}) x d x)$

Step 4

Multiply $4$ by $- 1$ .

$x^{2} (2 \sin (\frac{x}{2})) - 4 \int \sin (\frac{x}{2}) x d x$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sin (\frac{x}{2})$ .

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

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

Step 8

Multiply $- 2$ by $- 1$ .

$x^{2} (2 \sin (\frac{x}{2})) - 4 (x (- 2 \cos (\frac{x}{2})) + 2 \int \cos (\frac{x}{2}) d x)$

Step 9

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 9.1

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

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Step 9.1.1

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

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

Step 9.1.2

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 9.1.3

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 9.1.4

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

$\frac{1}{2}$

Step 9.2

Rewrite the problem using $u$ and $d u$ .

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

Step 10

Simplify.

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Step 10.1

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

$x^{2} (2 \sin (\frac{x}{2})) - 4 (x (- 2 \cos (\frac{x}{2})) + 2 \int \cos (u) (1 \cdot 2) d u)$

Step 10.2

Multiply $2$ by $1$ .

$x^{2} (2 \sin (\frac{x}{2})) - 4 (x (- 2 \cos (\frac{x}{2})) + 2 \int \cos (u) \cdot 2 d u)$

Step 10.3

Move $2$ to the left of $\cos (u)$ .

$x^{2} (2 \sin (\frac{x}{2})) - 4 (x (- 2 \cos (\frac{x}{2})) + 2 \int 2 \cos (u) d u)$

Step 11

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$x^{2} (2 \sin (\frac{x}{2})) - 4 (x (- 2 \cos (\frac{x}{2})) + 2 (2 \int \cos (u) d u))$

Step 12

Multiply $2$ by $2$ .

$x^{2} (2 \sin (\frac{x}{2})) - 4 (x (- 2 \cos (\frac{x}{2})) + 4 \int \cos (u) d u)$

Step 13

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

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

Step 14

Rewrite $x^{2} (2 \sin (\frac{x}{2})) - 4 (x (- 2 \cos (\frac{x}{2})) + 4 (\sin (u) + C))$ as $2 x^{2} \sin (\frac{x}{2}) - 4 (x (- 2 \cos (\frac{x}{2})) + 4 \sin (u)) + C$ .

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

Step 15

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

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

Step 16

Simplify.

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Step 16.1

Rewrite using the commutative property of multiplication.

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

Step 16.2

Apply the distributive property.

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

Step 16.3

Multiply $- 2$ by $- 4$ .

$2 x^{2} \sin (\frac{x}{2}) + 8 (x \cos (\frac{x}{2})) - 4 (4 \sin (\frac{x}{2})) + C$

Step 16.4

Multiply $4$ by $- 4$ .

$2 x^{2} \sin (\frac{x}{2}) + 8 x \cos (\frac{x}{2}) - 16 \sin (\frac{x}{2}) + C$

Step 17

Reorder terms.

$2 x^{2} \sin (\frac{1}{2} x) + 8 x \cos (\frac{1}{2} x) - 16 \sin (\frac{1}{2} x) + C$