Calculus Examples

$f (x) = - \frac{7}{x^{6}} + \frac{1}{3}$

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.

$F (x) = \int - \frac{7}{x^{6}} + \frac{1}{3} d x$

Step 3

Split the single integral into multiple integrals.

$\int - \frac{7}{x^{6}} d x + \int \frac{1}{3} d x$

Step 4

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

$- \int \frac{7}{x^{6}} d x + \int \frac{1}{3} d x$

Step 5

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$- (7 \int \frac{1}{x^{6}} d x) + \int \frac{1}{3} d x$

Step 6

Simplify the expression.

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Step 6.1

Multiply $7$ by $- 1$ .

$- 7 \int \frac{1}{x^{6}} d x + \int \frac{1}{3} d x$

Step 6.2

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

$- 7 \int {(x^{6})}^{- 1} d x + \int \frac{1}{3} d x$

Step 6.3

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

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Step 6.3.1

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

$- 7 \int x^{6 \cdot - 1} d x + \int \frac{1}{3} d x$

Step 6.3.2

Multiply $6$ by $- 1$ .

$- 7 \int x^{- 6} d x + \int \frac{1}{3} d x$

Step 7

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

$- 7 (- \frac{1}{5} x^{- 5} + C) + \int \frac{1}{3} d x$

Step 8

Simplify.

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Step 8.1

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

$- 7 (- \frac{x^{- 5}}{5} + C) + \int \frac{1}{3} d x$

Step 8.2

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

$- 7 (- \frac{1}{5 x^{5}} + C) + \int \frac{1}{3} d x$

Step 9

Apply the constant rule.

$- 7 (- \frac{1}{5 x^{5}} + C) + \frac{1}{3} x + C$

Step 10

Simplify.

$\frac{7}{5 x^{5}} + \frac{1}{3} x + C$

Step 11

The answer is the antiderivative of the function $f (x) = - \frac{7}{x^{6}} + \frac{1}{3}$ .

$F (x) =$ $\frac{7}{5 x^{5}} + \frac{1}{3} x + C$