Calculus Examples

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

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

$\cos (x)$ with respect to

$x$ is

$\sin (x)$ .

$\sin (x) + C$

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.

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

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.

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

Step 6

The integral of

$\sin (x)$ with respect to

$x$ is

$- \cos (x)$ .

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

Step 7

Apply the constant rule.

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

Step 8

Simplify.

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

Step 9

The function

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

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

Step 10

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 11

Split the single integral into multiple integrals.

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

Step 12

Since

$- 1$ is constant with respect to

$x$ , move

$- 1$ out of the integral.

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

Step 13

The integral of

$\cos (x)$ with respect to

$x$ is

$\sin (x)$ .

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

Step 14

Since

$C$ is constant with respect to

$x$ , move

$C$ out of the integral.

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

Step 15

By the Power Rule, the integral of

$x$ with respect to

$x$ is

$\frac{1}{2} x^{2}$ .

$- (\sin (x) + C) + C (\frac{1}{2} x^{2} + C) + \int C d x$

Step 16

Apply the constant rule.

$- (\sin (x) + C) + C (\frac{1}{2} x^{2} + C) + C x + C$

Step 17

Simplify.

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Step 17.1

Combine

$\frac{1}{2}$ and

$x^{2}$ .

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

Step 17.2

Simplify.

$- \sin (x) + \frac{1}{2} C x^{2} + C x + C$

Step 18

The function

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

$f (x) = - \sin (x) + \frac{1}{2} \cdot (C x^{2}) + C x + C$