Calculus Examples

$x \cdot \cos (x)$

Step 1

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

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

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.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

The answer is the antiderivative of the function $f (x) = x \cdot \cos (x)$ .

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