Calculus Examples

$x^{3} \sin (x)$

Step 1

Write $x^{3} \sin (x)$ as a function.

$f (x) = x^{3} \sin (x)$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int x^{3} \sin (x) d x$

Step 4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{3}$ and $d v = \sin (x)$ .

$x^{3} (- \cos (x)) - \int - \cos (x) (3 x^{2}) d x$

Step 5

Multiply $3$ by $- 1$ .

$x^{3} (- \cos (x)) - \int - 3 \cos (x) x^{2} d x$

Step 6

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$x^{3} (- \cos (x)) - (- 3 \int \cos (x) x^{2} d x)$

Step 7

Multiply $- 3$ by $- 1$ .

$x^{3} (- \cos (x)) + 3 \int \cos (x) x^{2} d x$

Step 8

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

$x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - \int \sin (x) (2 x) d x)$

Step 9

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

$x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - (2 \int \sin (x) (x) d x))$

Step 10

Multiply $2$ by $- 1$ .

$x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - 2 \int \sin (x) (x) d x)$

Step 11

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

$x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - 2 (x (- \cos (x)) - \int - \cos (x) d x))$

Step 12

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

$x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - 2 (x (- \cos (x)) - - \int \cos (x) d x))$

Step 13

Simplify.

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Step 13.1

Multiply $- 1$ by $- 1$ .

$x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - 2 (x (- \cos (x)) + 1 \int \cos (x) d x))$

Step 13.2

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

$x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - 2 (x (- \cos (x)) + \int \cos (x) d x))$

Step 14

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

$x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - 2 (x (- \cos (x)) + \sin (x) + C))$

Step 15

Rewrite $x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - 2 (x (- \cos (x)) + \sin (x) + C))$ as $- x^{3} \cos (x) + 3 (x^{2} \sin (x) - 2 (x (- \cos (x)) + \sin (x))) + C$ .

$- x^{3} \cos (x) + 3 (x^{2} \sin (x) - 2 (x (- \cos (x)) + \sin (x))) + C$

Step 16

The answer is the antiderivative of the function $f (x) = x^{3} \sin (x)$ .

$F (x) =$ $- x^{3} \cos (x) + 3 (x^{2} \sin (x) - 2 (x (- \cos (x)) + \sin (x))) + C$