Calculus Examples

x \cdot \cos (x)

$x \cdot \cos (x)$

Step 1

Write

x \cdot \cos (x)

$x \cdot \cos (x)$ as a function.

f (x) = x \cdot \cos (x)

$f (x) = x \cdot \cos (x)$

Step 2

The function

F (x)

$F (x)$ can be found by finding the indefinite integral of the derivative

f (x)

$f (x)$ .

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

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

Step 3

Set up the integral to solve.

F (x) = \int x \cdot \cos (x) d x

$F (x) = \int x \cdot \cos (x) d x$

Step 4

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

$u = x$ and

d v = \cos (x)

$d v = \cos (x)$ .

x \sin (x) - \int \sin (x) d x

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

Step 5

The integral of

\sin (x)

$\sin (x)$ with respect to

x

$x$ is

- \cos (x)

$- \cos (x)$ .

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

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

Step 6

Rewrite

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

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

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

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

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

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

Step 7

The answer is the antiderivative of the function

f (x) = x \cdot \cos (x)

$f (x) = x \cdot \cos (x)$ .

F (x) =

$F (x) =$

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

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