Calculus Examples

$\int u^{2} \sin (2 u) d u$

Step 1

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Multiply $2$ by $- 1$ .

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

Step 4.2

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

$- \frac{u^{2} \cos (2 u)}{2} - (\frac{- 2}{2} \int \cos (2 u) (u) d u)$

Step 4.3

Cancel the common factor of $- 2$ and $2$ .

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Step 4.3.1

Factor $2$ out of $- 2$ .

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

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $2$ out of $2$ .

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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

Step 4.3.2.4

Divide $- 1$ by $1$ .

$- \frac{u^{2} \cos (2 u)}{2} - - \int \cos (2 u) (u) d u$

Step 4.4

Multiply $- 1$ by $- 1$ .

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

Step 4.5

Multiply $\int \cos (2 u) (u) d u$ by $1$ .

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

Step 5

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = u$ and $dv = \cos (2 u)$ .

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Combine $u$ and $\frac{\sin (2 u)}{2}$ .

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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Step 8.1.1

Differentiate $2 u$ .

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

Step 8.1.2

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

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

Step 8.1.3

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

$2 \cdot 1$

Step 8.1.4

Multiply $2$ by $1$ .

$2$

Step 8.2

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

Multiply $2$ by $2$ .

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

Step 12

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

$- \frac{u^{2} \cos (2 u)}{2} + \frac{u \sin (2 u)}{2} - \frac{1}{4} (- \cos (u_{1}) + C)$

Step 13

Rewrite $- \frac{u^{2} \cos (2 u)}{2} + \frac{u \sin (2 u)}{2} - \frac{1}{4} (- \cos (u_{1}) + C)$ as $- \frac{u^{2} \cos (2 u)}{2} + \frac{u \sin (2 u)}{2} + \frac{\cos (u_{1})}{4} + C$ .

$- \frac{u^{2} \cos (2 u)}{2} + \frac{u \sin (2 u)}{2} + \frac{\cos (u_{1})}{4} + C$

Step 14

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

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

Step 15

Reorder terms.

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