Calculus Examples

$f (x) = \frac{8}{x^{7}}$

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.

$F (x) = \int \frac{8}{x^{7}} d x$

Step 3

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int \frac{1}{x^{7}} d x$

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $x^{7}$ out of the denominator by raising it to the $- 1$ power.

$8 \int {(x^{7})}^{- 1} d x$

Step 4.2

Multiply the exponents in ${(x^{7})}^{- 1}$ .

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Step 4.2.1

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

$8 \int x^{7 \cdot - 1} d x$

Step 4.2.2

Multiply $7$ by $- 1$ .

$8 \int x^{- 7} d x$

Step 5

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

$8 (- \frac{1}{6} x^{- 6} + C)$

Step 6

Simplify the answer.

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Step 6.1

Simplify.

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Step 6.1.1

Combine $x^{- 6}$ and $\frac{1}{6}$ .

$8 (- \frac{x^{- 6}}{6} + C)$

Step 6.1.2

Move $x^{- 6}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$8 (- \frac{1}{6 x^{6}} + C)$

Step 6.2

Simplify.

$8 (- \frac{1}{6 x^{6}}) + C$

Step 6.3

Simplify.

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Step 6.3.1

Multiply $- 1$ by $8$ .

$- 8 \frac{1}{6 x^{6}} + C$

Step 6.3.2

Combine $- 8$ and $\frac{1}{6 x^{6}}$ .

$\frac{- 8}{6 x^{6}} + C$

Step 6.3.3

Cancel the common factor of $- 8$ and $6$ .

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Step 6.3.3.1

Factor $2$ out of $- 8$ .

$\frac{2 (- 4)}{6 x^{6}} + C$

Step 6.3.3.2

Cancel the common factors.

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Step 6.3.3.2.1

Factor $2$ out of $6 x^{6}$ .

$\frac{2 (- 4)}{2 (3 x^{6})} + C$

Step 6.3.3.2.2

Cancel the common factor.

$\frac{2 \cdot - 4}{2 (3 x^{6})} + C$

Step 6.3.3.2.3

Rewrite the expression.

$\frac{- 4}{3 x^{6}} + C$

Step 6.3.4

Move the negative in front of the fraction.

$- \frac{4}{3 x^{6}} + C$

Step 7

The answer is the antiderivative of the function $f (x) = \frac{8}{x^{7}}$ .

$F (x) =$ $- \frac{4}{3 x^{6}} + C$