Calculus Examples

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

Step 1

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

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

Step 2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 3

Simplify.

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Step 3.1

Multiply $- 1$ by $- 1$ .

$x (- \cos (x)) + 1 \int \cos (x) d x$

Step 3.2

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

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

Step 4

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

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

Step 5

Rewrite $x (- \cos (x)) + \sin (x) + C$ as $- x \cos (x) + \sin (x) + C$ .

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