Calculus Examples

$8 x^{9} - 2 x^{6} + 13 x^{3}$

Step 1

Write $8 x^{9} - 2 x^{6} + 13 x^{3}$ as a function.

$f (x) = 8 x^{9} - 2 x^{6} + 13 x^{3}$

Step 2

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

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

Step 3

Set up the integral to solve.

$F (x) = \int 8 x^{9} - 2 x^{6} + 13 x^{3} d x$

Step 4

Split the single integral into multiple integrals.

$\int 8 x^{9} d x + \int - 2 x^{6} d x + \int 13 x^{3} d x$

Step 5

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int x^{9} d x + \int - 2 x^{6} d x + \int 13 x^{3} d x$

Step 6

By the Power Rule, the integral of $x^{9}$ with respect to $x$ is $\frac{1}{10} x^{10}$ .

$8 (\frac{1}{10} x^{10} + C) + \int - 2 x^{6} d x + \int 13 x^{3} d x$

Step 7

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

$8 (\frac{1}{10} x^{10} + C) - 2 \int x^{6} d x + \int 13 x^{3} d x$

Step 8

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

$8 (\frac{1}{10} x^{10} + C) - 2 (\frac{1}{7} x^{7} + C) + \int 13 x^{3} d x$

Step 9

Since $13$ is constant with respect to $x$ , move $13$ out of the integral.

$8 (\frac{1}{10} x^{10} + C) - 2 (\frac{1}{7} x^{7} + C) + 13 \int x^{3} d x$

Step 10

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$8 (\frac{1}{10} x^{10} + C) - 2 (\frac{1}{7} x^{7} + C) + 13 (\frac{1}{4} x^{4} + C)$

Step 11

Simplify.

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Step 11.1

Simplify.

$\frac{4 x^{10}}{5} - \frac{2 x^{7}}{7} + 13 (\frac{1}{4} x^{4}) + C$

Step 11.2

Simplify.

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Step 11.2.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$\frac{4 x^{10}}{5} - \frac{2 x^{7}}{7} + 13 \frac{x^{4}}{4} + C$

Step 11.2.2

Combine $13$ and $\frac{x^{4}}{4}$ .

$\frac{4 x^{10}}{5} - \frac{2 x^{7}}{7} + \frac{13 x^{4}}{4} + C$

Step 12

Reorder terms.

$\frac{4}{5} x^{10} - \frac{2}{7} x^{7} + \frac{13}{4} x^{4} + C$

Step 13

The answer is the antiderivative of the function $f (x) = 8 x^{9} - 2 x^{6} + 13 x^{3}$ .

$F (x) =$ $\frac{4}{5} x^{10} - \frac{2}{7} x^{7} + \frac{13}{4} x^{4} + C$