Calculus Examples

$f (x) = \sqrt[5]{39 x^{7}}$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int \sqrt[5]{39 x^{7}} d x$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Rewrite $39 x^{7}$ as $x^{5} \cdot (39 x^{2})$ .

Tap for more steps...

Step 3.1.1

Factor out $x^{5}$ .

$\int \sqrt[5]{39 (x^{5} x^{2})} d x$

Step 3.1.2

Reorder $39$ and $x^{5}$ .

$\int \sqrt[5]{x^{5} \cdot 39 x^{2}} d x$

Step 3.1.3

Add parentheses.

$\int \sqrt[5]{x^{5} \cdot (39 x^{2})} d x$

Step 3.2

Pull terms out from under the radical.

$\int x \sqrt[5]{39 x^{2}} d x$

Step 4

Let $u = 39 x^{2}$ . Then $d u = 78 x d x$ , so $\frac{1}{78} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 4.1

Let $u = 39 x^{2}$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 4.1.1

Differentiate $39 x^{2}$ .

$\frac{d}{d x} [39 x^{2}]$

Step 4.1.2

Since $39$ is constant with respect to $x$ , the derivative of $39 x^{2}$ with respect to $x$ is $39 \frac{d}{d x} [x^{2}]$ .

$39 \frac{d}{d x} [x^{2}]$

Step 4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$39 (2 x)$

Step 4.1.4

Multiply $2$ by $39$ .

$78 x$

Step 4.2

Rewrite the problem using $u$ and $d u$ .

$\int \sqrt[5]{u} \frac{1}{78} d u$

Step 5

Combine $\sqrt[5]{u}$ and $\frac{1}{78}$ .

$\int \frac{\sqrt[5]{u}}{78} d u$

Step 6

Since $\frac{1}{78}$ is constant with respect to $u$ , move $\frac{1}{78}$ out of the integral.

$\frac{1}{78} \int \sqrt[5]{u} d u$

Step 7

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{u}$ as $u^{\frac{1}{5}}$ .

$\frac{1}{78} \int u^{\frac{1}{5}} d u$

Step 8

By the Power Rule, the integral of $u^{\frac{1}{5}}$ with respect to $u$ is $\frac{5}{6} u^{\frac{6}{5}}$ .

$\frac{1}{78} (\frac{5}{6} u^{\frac{6}{5}} + C)$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Rewrite $\frac{1}{78} (\frac{5}{6} u^{\frac{6}{5}} + C)$ as $\frac{1}{78} \cdot \frac{5}{6} u^{\frac{6}{5}} + C$ .

$\frac{1}{78} \cdot \frac{5}{6} u^{\frac{6}{5}} + C$

Step 9.2

Simplify.

Tap for more steps...

Step 9.2.1

Multiply $\frac{1}{78}$ by $\frac{5}{6}$ .

$\frac{5}{78 \cdot 6} u^{\frac{6}{5}} + C$

Step 9.2.2

Multiply $78$ by $6$ .

$\frac{5}{468} u^{\frac{6}{5}} + C$

Step 10

Replace all occurrences of $u$ with $39 x^{2}$ .

$\frac{5}{468} {(39 x^{2})}^{\frac{6}{5}} + C$

Step 11

The answer is the antiderivative of the function $f (x) = \sqrt[5]{39 x^{7}}$ .

$F (x) =$ $\frac{5}{468} {(39 x^{2})}^{\frac{6}{5}} + C$