Calculus Examples

$\int (x^{8} + x^{6} - x^{2}) d x$

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

By the Power Rule, the integral of $x^{8}$ with respect to $x$ is $\frac{1}{9} x^{9}$ .

$\frac{1}{9} x^{9} + C + \int x^{6} d x + \int - x^{2} d x$

Step 4

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

$\frac{1}{9} x^{9} + C + \frac{1}{7} x^{7} + C + \int - x^{2} d x$

Step 5

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

$\frac{1}{9} x^{9} + C + \frac{1}{7} x^{7} + C - \int x^{2} d x$

Step 6

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

$\frac{1}{9} x^{9} + C + \frac{1}{7} x^{7} + C - (\frac{1}{3} x^{3} + C)$

Step 7

Simplify.

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Step 7.1

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

$\frac{1}{9} x^{9} + C + \frac{1}{7} x^{7} + C - (\frac{x^{3}}{3} + C)$

Step 7.2

Simplify.

$\frac{1}{9} x^{9} + \frac{1}{7} x^{7} - \frac{1}{3} x^{3} + C$