Calculus Examples

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

Step 1

Integrate by parts using the formula

$\int u d v = u v - \int v d u$ , where

$u = x^{2}$ and

$d v = \sin (x)$ .

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

Step 2

Multiply

$2$ by

$- 1$ .

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

Step 3

Since

$- 2$ is constant with respect to

$x$ , move

$- 2$ out of the integral.

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

Step 4

Multiply

$- 2$ by

$- 1$ .

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

Step 5

Integrate by parts using the formula

$\int u d v = u v - \int v d u$ , where

$u = x$ and

$d v = \cos (x)$ .

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

Step 6

The integral of

$\sin (x)$ with respect to

$x$ is

$- \cos (x)$ .

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

Step 7

Simplify the answer.

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Step 7.1

Rewrite

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

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

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

Step 7.2

Simplify.

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Step 7.2.1

Multiply

$- 1$ by

$- 1$ .

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

Step 7.2.2

Multiply

$\cos (x)$ by

$1$ .

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