Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Multiply $2$ by $- 1$ .

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

Step 4

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

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

Step 5

Multiply $- 2$ by $- 1$ .

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

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

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

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

Step 7.2

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

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