Calculus Examples

$x^{6} - \frac{1}{x^{6}}$

Step 1

Write $x^{6} - \frac{1}{x^{6}}$ as a function.

$f (x) = x^{6} - \frac{1}{x^{6}}$

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.

$F (x) = \int x^{6} - \frac{1}{x^{6}} d x$

Step 4

Split the single integral into multiple integrals.

$\int x^{6} d x + \int - \frac{1}{x^{6}} d x$

Step 5

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$\frac{1}{7} x^{7} + C + \int - \frac{1}{x^{6}} d x$

Step 6

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{7} x^{7} + C - \int \frac{1}{x^{6}} d x$

Step 7

Apply basic rules of exponents.

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Step 7.1

Move $x^{6}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{7} x^{7} + C - \int {(x^{6})}^{- 1} d x$

Step 7.2

Multiply the exponents in ${(x^{6})}^{- 1}$ .

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Step 7.2.1

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

$\frac{1}{7} x^{7} + C - \int x^{6 \cdot - 1} d x$

Step 7.2.2

Multiply $6$ by $- 1$ .

$\frac{1}{7} x^{7} + C - \int x^{- 6} d x$

Step 8

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

$\frac{1}{7} x^{7} + C - (- \frac{1}{5} x^{- 5} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

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Step 9.1.1

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

$\frac{1}{7} x^{7} + C - (- \frac{x^{- 5}}{5} + C)$

Step 9.1.2

Move $x^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{7} x^{7} + C - (- \frac{1}{5 x^{5}} + C)$

Step 9.2

Simplify.

$\frac{1}{7} x^{7} - - \frac{1}{5 x^{5}} + C$

Step 9.3

Simplify.

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Step 9.3.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{7} x^{7} + 1 \frac{1}{5 x^{5}} + C$

Step 9.3.2

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

$\frac{1}{7} x^{7} + \frac{1}{5 x^{5}} + C$

Step 10

The answer is the antiderivative of the function $f (x) = x^{6} - \frac{1}{x^{6}}$ .

$F (x) =$ $\frac{1}{7} x^{7} + \frac{1}{5 x^{5}} + C$