Calculus Examples

$\int (\sqrt{x^{4}} - \sqrt[5]{x^{8}}) d x$

Step 1

Remove parentheses.

$\int \sqrt{x^{4}} - \sqrt[5]{x^{8}} d x$

Step 2

Split the single integral into multiple integrals.

$\int \sqrt{x^{4}} d x + \int - \sqrt[5]{x^{8}} d x$

Step 3

Simplify the expression.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{4}}$ as $x^{\frac{4}{2}}$ .

$\int x^{\frac{4}{2}} d x + \int - \sqrt[5]{x^{8}} d x$

Step 3.2

Divide $4$ by $2$ .

$\int x^{2} d x + \int - \sqrt[5]{x^{8}} d x$

Step 4

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

$\frac{1}{3} x^{3} + C + \int - \sqrt[5]{x^{8}} d x$

Step 5

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

$\frac{1}{3} x^{3} + C - \int \sqrt[5]{x^{8}} d x$

Step 6

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{8}}$ as $x^{\frac{8}{5}}$ .

$\frac{1}{3} x^{3} + C - \int x^{\frac{8}{5}} d x$

Step 7

By the Power Rule, the integral of $x^{\frac{8}{5}}$ with respect to $x$ is $\frac{5}{13} x^{\frac{13}{5}}$ .

$\frac{1}{3} x^{3} + C - (\frac{5}{13} x^{\frac{13}{5}} + C)$

Step 8

Simplify.

$\frac{1}{3} x^{3} - \frac{5}{13} x^{\frac{13}{5}} + C$