Calculus Examples

$\frac{6}{5} \cdot \sqrt[5]{x}$

Step 1

Write $\frac{6}{5} \cdot \sqrt[5]{x}$ as a function.

$f (x) = \frac{6}{5} \cdot \sqrt[5]{x}$

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.

$F (x) = \int \frac{6}{5} \cdot \sqrt[5]{x} d x$

Step 4

Since $\frac{6}{5}$ is constant with respect to $x$ , move $\frac{6}{5}$ out of the integral.

$\frac{6}{5} \int \sqrt[5]{x} d x$

Step 5

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

$\frac{6}{5} \int x^{\frac{1}{5}} d x$

Step 6

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

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

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

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

$\frac{6}{5} \cdot \frac{5}{6} x^{\frac{6}{5}} + C$

Step 7.2

Simplify.

Tap for more steps...

Step 7.2.1

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

$\frac{6 \cdot 5}{5 \cdot 6} x^{\frac{6}{5}} + C$

Step 7.2.2

Multiply $6$ by $5$ .

$\frac{30}{5 \cdot 6} x^{\frac{6}{5}} + C$

Step 7.2.3

Multiply $5$ by $6$ .

$\frac{30}{30} x^{\frac{6}{5}} + C$

Step 7.2.4

Cancel the common factor of $30$ .

Tap for more steps...

Step 7.2.4.1

Cancel the common factor.

$\frac{30}{30} x^{\frac{6}{5}} + C$

Step 7.2.4.2

Rewrite the expression.

$1 x^{\frac{6}{5}} + C$

Step 7.2.5

Multiply $x^{\frac{6}{5}}$ by $1$ .

$x^{\frac{6}{5}} + C$

Step 8

The answer is the antiderivative of the function $f (x) = \frac{6}{5} \cdot \sqrt[5]{x}$ .

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