Calculus Examples

$\int (2 \sqrt[6]{x^{4}} - 18) d x$

Step 1

Remove parentheses.

$\int 2 \sqrt[6]{x^{4}} - 18 d x$

Step 2

Split the single integral into multiple integrals.

$\int 2 \sqrt[6]{x^{4}} d x + \int - 18 d x$

Step 3

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

$2 \int \sqrt[6]{x^{4}} d x + \int - 18 d x$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x^{4}}$ as $x^{\frac{4}{6}}$ .

$2 \int x^{\frac{4}{6}} d x + \int - 18 d x$

Step 4.2

Cancel the common factor of $4$ and $6$ .

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Step 4.2.1

Factor $2$ out of $4$ .

$2 \int x^{\frac{2 (2)}{6}} d x + \int - 18 d x$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $2$ out of $6$ .

$2 \int x^{\frac{2 \cdot 2}{2 \cdot 3}} d x + \int - 18 d x$

Step 4.2.2.2

Cancel the common factor.

$2 \int x^{\frac{2 \cdot 2}{2 \cdot 3}} d x + \int - 18 d x$

Step 4.2.2.3

Rewrite the expression.

$2 \int x^{\frac{2}{3}} d x + \int - 18 d x$

Step 5

By the Power Rule, the integral of $x^{\frac{2}{3}}$ with respect to $x$ is $\frac{3}{5} x^{\frac{5}{3}}$ .

$2 (\frac{3}{5} x^{\frac{5}{3}} + C) + \int - 18 d x$

Step 6

Apply the constant rule.

$2 (\frac{3}{5} x^{\frac{5}{3}} + C) - 18 x + C$

Step 7

Simplify.

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Step 7.1

Combine $\frac{3}{5}$ and $x^{\frac{5}{3}}$ .

$2 (\frac{3 x^{\frac{5}{3}}}{5} + C) - 18 x + C$

Step 7.2

Simplify.

$\frac{6 x^{\frac{5}{3}}}{5} - 18 x + C$

Step 7.3

Reorder terms.

$\frac{6}{5} x^{\frac{5}{3}} - 18 x + C$