Calculus Examples

$x^{2} \sin (2 x)$

Step 1

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Multiply $2$ by $- 1$ .

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

Factor $2$ out of $- 2$ .

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

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $2$ out of $2$ .

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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

Step 4.3.2.4

Divide $- 1$ by $1$ .

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

Step 4.4

Multiply $- 1$ by $- 1$ .

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

Step 4.5

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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Step 8.1.1

Differentiate $2 x$ .

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

Step 8.1.2

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

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

Step 8.1.3

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

$2 \cdot 1$

Step 8.1.4

Multiply $2$ by $1$ .

$2$

Step 8.2

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

Multiply $2$ by $2$ .

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

Step 12

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

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

Step 13

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

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

Step 14

Replace all occurrences of $u$ with $2 x$ .

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

Step 15

Reorder terms.

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