Calculus Examples

$\int 2 x \sin (x) d x$

Step 1

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

$2 x (- \cos (x)) - \int - \cos (x) \cdot 2 d x$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Multiply $- 1$ by $2$ .

$- 2 x \cos (x) - \int - \cos (x) \cdot 2 d x$

Step 2.2

Multiply $2$ by $- 1$ .

$- 2 x \cos (x) - \int - 2 \cos (x) d x$

Step 3

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

$- 2 x \cos (x) - (- 2 \int \cos (x) d x)$

Step 4

Multiply $- 2$ by $- 1$ .

$- 2 x \cos (x) + 2 \int \cos (x) d x$

Step 5

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

$- 2 x \cos (x) + 2 (\sin (x) + C)$

Step 6

Simplify.

$- 2 x \cos (x) + 2 \sin (x) + C$