Calculus Examples

$x^{- \frac{6}{5}}$

Step 1

Write $x^{- \frac{6}{5}}$ as a function.

$f (x) = x^{- \frac{6}{5}}$

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.

$F (x) = \int x^{- \frac{6}{5}} d x$

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

Rewrite $- 5 x^{- \frac{1}{5}} + C$ as $- 5 \frac{1}{x^{\frac{1}{5}}} + C$ .

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

Step 5.2

Simplify.

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Step 5.2.1

Combine $- 5$ and $\frac{1}{x^{\frac{1}{5}}}$ .

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

Step 5.2.2

Move the negative in front of the fraction.

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

Step 6

The answer is the antiderivative of the function $f (x) = x^{- \frac{6}{5}}$ .

$F (x) =$ $- \frac{5}{x^{\frac{1}{5}}} + C$