Calculus Examples

$\int 3 x^{2} \sin (x) d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int x^{2} \sin (x) d x$

Step 2

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

$3 (x^{2} (- \cos (x)) - \int - \cos (x) (2 x) d x)$

Step 3

Multiply $2$ by $- 1$ .

$3 (x^{2} (- \cos (x)) - \int - 2 \cos (x) x d x)$

Step 4

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

$3 (x^{2} (- \cos (x)) - (- 2 \int \cos (x) x d x))$

Step 5

Multiply $- 2$ by $- 1$ .

$3 (x^{2} (- \cos (x)) + 2 \int \cos (x) x d x)$

Step 6

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

$3 (x^{2} (- \cos (x)) + 2 (x \sin (x) - \int \sin (x) d x))$

Step 7

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

$3 (x^{2} (- \cos (x)) + 2 (x \sin (x) - (- \cos (x) + C)))$

Step 8

Rewrite $3 (x^{2} (- \cos (x)) + 2 (x \sin (x) - (- \cos (x) + C)))$ as $3 (- x^{2} \cos (x) + 2 x \sin (x) + 2 \cos (x)) + C$ .

$3 (- x^{2} \cos (x) + 2 x \sin (x) + 2 \cos (x)) + C$