Calculus Examples

${(\sqrt[6]{x})}^{3}$

Step 1

Write ${(\sqrt[6]{x})}^{3}$ as a function.

$f (x) = {(\sqrt[6]{x})}^{3}$

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.

$F (x) = \int {(\sqrt[6]{x})}^{3} d x$

Step 4

Simplify by cancelling exponent with radical.

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Step 4.1

Rewrite ${(\sqrt[6]{x})}^{3}$ as $\sqrt{x}$ .

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x}$ as $x^{\frac{1}{6}}$ .

$\int {(x^{\frac{1}{6}})}^{3} d x$

Step 4.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int x^{\frac{1}{6} \cdot 3} d x$

Step 4.1.3

Combine $\frac{1}{6}$ and $3$ .

$\int x^{\frac{3}{6}} d x$

Step 4.1.4

Cancel the common factor of $3$ and $6$ .

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Step 4.1.4.1

Factor $3$ out of $3$ .

$\int x^{\frac{3 (1)}{6}} d x$

Step 4.1.4.2

Cancel the common factors.

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Step 4.1.4.2.1

Factor $3$ out of $6$ .

$\int x^{\frac{3 \cdot 1}{3 \cdot 2}} d x$

Step 4.1.4.2.2

Cancel the common factor.

$\int x^{\frac{3 \cdot 1}{3 \cdot 2}} d x$

Step 4.1.4.2.3

Rewrite the expression.

$\int x^{\frac{1}{2}} d x$

Step 4.1.5

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$\int \sqrt{x} d x$

Step 4.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\int x^{\frac{1}{2}} d x$

Step 5

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

$\frac{2}{3} x^{\frac{3}{2}} + C$

Step 6

The answer is the antiderivative of the function $f (x) = {(\sqrt[6]{x})}^{3}$ .

$F (x) =$ $\frac{2}{3} x^{\frac{3}{2}} + C$