Calculus Examples

$f'' (x) = \sin (x)$

Step 1

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

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

Step 2

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

$- \cos (x) + C$

Step 3

The function $f'$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$f' (x) = - \cos (x) + C$

Step 4

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

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

Step 5

Split the single integral into multiple integrals.

$\int - \cos (x) d x + \int C d x$

Step 6

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

$- \int \cos (x) d x + \int C d x$

Step 7

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

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

Step 8

Apply the constant rule.

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

Step 9

Simplify.

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

Step 10

The function $f$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$f (x) = - \sin (x) + C x + C$