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.

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