Calculus Examples

$f'' (x) = x^{6} - 4 x^{4} + x + 1$

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.

$\int x^{6} d x + \int - 4 x^{4} d x + \int x d x + \int d x$

Step 3

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$\frac{1}{7} x^{7} + C + \int - 4 x^{4} d x + \int x d x + \int d x$

Step 4

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

$\frac{1}{7} x^{7} + C - 4 \int x^{4} d x + \int x d x + \int d x$

Step 5

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$\frac{1}{7} x^{7} + C - 4 (\frac{1}{5} x^{5} + C) + \int x d x + \int d x$

Step 6

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

$\frac{1}{7} x^{7} + C - 4 (\frac{1}{5} x^{5} + C) + \frac{1}{2} x^{2} + C + \int d x$

Step 7

Combine $\frac{1}{5}$ and $x^{5}$ .

$\frac{1}{7} x^{7} + C - 4 (\frac{x^{5}}{5} + C) + \frac{1}{2} x^{2} + C + \int d x$

Step 8

Apply the constant rule.

$\frac{1}{7} x^{7} + C - 4 (\frac{x^{5}}{5} + C) + \frac{1}{2} x^{2} + C + x + C$

Step 9

Simplify.

$\frac{x^{7}}{7} - \frac{4 x^{5}}{5} + \frac{x^{2}}{2} + x + C$

Step 10

Reorder terms.

$\frac{1}{7} x^{7} - \frac{4}{5} x^{5} + \frac{1}{2} x^{2} + x + C$

Step 11

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

$f' (x) = \frac{1}{7} x^{7} - \frac{4}{5} x^{5} + \frac{1}{2} x^{2} + x + C$

Step 12

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 13

Split the single integral into multiple integrals.

$\int \frac{1}{7} x^{7} d x + \int - \frac{4}{5} x^{5} d x + \int \frac{1}{2} x^{2} d x + \int x d x + \int C d x$

Step 14

Since $\frac{1}{7}$ is constant with respect to $x$ , move $\frac{1}{7}$ out of the integral.

$\frac{1}{7} \int x^{7} d x + \int - \frac{4}{5} x^{5} d x + \int \frac{1}{2} x^{2} d x + \int x d x + \int C d x$

Step 15

By the Power Rule, the integral of $x^{7}$ with respect to $x$ is $\frac{1}{8} x^{8}$ .

$\frac{1}{7} (\frac{1}{8} x^{8} + C) + \int - \frac{4}{5} x^{5} d x + \int \frac{1}{2} x^{2} d x + \int x d x + \int C d x$

Step 16

Since $- \frac{4}{5}$ is constant with respect to $x$ , move $- \frac{4}{5}$ out of the integral.

$\frac{1}{7} (\frac{1}{8} x^{8} + C) - \frac{4}{5} \int x^{5} d x + \int \frac{1}{2} x^{2} d x + \int x d x + \int C d x$

Step 17

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$\frac{1}{7} (\frac{1}{8} x^{8} + C) - \frac{4}{5} (\frac{1}{6} x^{6} + C) + \int \frac{1}{2} x^{2} d x + \int x d x + \int C d x$

Step 18

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{7} (\frac{1}{8} x^{8} + C) - \frac{4}{5} (\frac{1}{6} x^{6} + C) + \frac{1}{2} \int x^{2} d x + \int x d x + \int C d x$

Step 19

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

$\frac{1}{7} (\frac{1}{8} x^{8} + C) - \frac{4}{5} (\frac{1}{6} x^{6} + C) + \frac{1}{2} (\frac{1}{3} x^{3} + C) + \int x d x + \int C d x$

Step 20

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

$\frac{1}{7} (\frac{1}{8} x^{8} + C) - \frac{4}{5} (\frac{1}{6} x^{6} + C) + \frac{1}{2} (\frac{1}{3} x^{3} + C) + \frac{1}{2} x^{2} + C + \int C d x$

Step 21

Apply the constant rule.

$\frac{1}{7} (\frac{1}{8} x^{8} + C) - \frac{4}{5} (\frac{1}{6} x^{6} + C) + \frac{1}{2} (\frac{1}{3} x^{3} + C) + \frac{1}{2} x^{2} + C + C x + C$

Step 22

Simplify.

$\frac{x^{8}}{56} - \frac{2 x^{6}}{15} + \frac{x^{3}}{6} + \frac{x^{2}}{2} + C x + C$

Step 23

Reorder terms.

$\frac{1}{56} x^{8} - \frac{2}{15} x^{6} + \frac{1}{6} x^{3} + \frac{1}{2} x^{2} + C x + C$

Step 24

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

$f (x) = \frac{1}{56} x^{8} - \frac{2}{15} x^{6} + \frac{1}{6} x^{3} + \frac{1}{2} x^{2} + C x + C$