Calculus Examples

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

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

Step 1

Integrate by parts using the formula

\int u d v = u v - \int v d u

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

u = x^{2} + 2 x

$u = x^{2} + 2 x$ and

d v = cos (x)

$d v = \cos (x)$ .

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

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

Step 2

Integrate by parts using the formula

\int u d v = u v - \int v d u

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

u = 2 x + 2

$u = 2 x + 2$ and

d v = sin (x)

$d v = \sin (x)$ .

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

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

Step 3

Multiply

2

$2$ by

- 1

$- 1$ .

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

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

Step 4

Since

- 2

$- 2$ is constant with respect to

x

$x$ , move

- 2

$- 2$ out of the integral.

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

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

Step 5

Multiply

- 2

$- 2$ by

- 1

$- 1$ .

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

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

Step 6

The integral of

cos (x)

$\cos (x)$ with respect to

x

$x$ is

sin (x)

$\sin (x)$ .

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

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

Step 7

Rewrite

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

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

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

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

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

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