Calculus Examples

$f'' (x) = 4 x + \sin (x)$

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

Split the single integral into multiple integrals.

$\int 4 x d x + \int \sin (x) d x$

Step 3

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int x d x + \int \sin (x) d x$

Step 4

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$4 (\frac{1}{2} x^{2} + C) + \int \sin (x) d x$

Step 5

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

$4 (\frac{1}{2} x^{2} + C) - \cos (x) + C$

Step 6

Simplify.

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Step 6.1

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

$4 (\frac{x^{2}}{2} + C) - \cos (x) + C$

Step 6.2

Simplify.

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

Step 7

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

$f' (x) = 2 x^{2} - \cos (x) + C$

Step 8

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 9

Split the single integral into multiple integrals.

$\int 2 x^{2} d x + \int - \cos (x) d x + \int C d x$

Step 10

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int x^{2} d x + \int - \cos (x) d x + \int C d x$

Step 11

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$2 (\frac{1}{3} x^{3} + C) + \int - \cos (x) d x + \int C d x$

Step 12

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

$2 (\frac{1}{3} x^{3} + C) - \int \cos (x) d x + \int C d x$

Step 13

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

$2 (\frac{1}{3} x^{3} + C) - (\sin (x) + C) + \int C d x$

Step 14

Apply the constant rule.

$2 (\frac{1}{3} x^{3} + C) - (\sin (x) + C) + C x + C$

Step 15

Simplify.

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Step 15.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$2 (\frac{x^{3}}{3} + C) - (\sin (x) + C) + C x + C$

Step 15.2

Simplify.

$\frac{2 x^{3}}{3} - \sin (x) + C x + C$

Step 15.3

Reorder terms.

$\frac{2}{3} x^{3} - \sin (x) + C x + C$

Step 16

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

$f (x) = \frac{2}{3} x^{3} - \sin (x) + C x + C$