Calculus Examples

$\int y \sin (y) d y$

Step 1

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

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2

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

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

Step 4

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

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

Step 5

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

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