Calculus Examples

$- 2 x^{- 3} + 20 x^{- 5}$

Step 1

Write $- 2 x^{- 3} + 20 x^{- 5}$ as a function.

$f (x) = - 2 x^{- 3} + 20 x^{- 5}$

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.

$F (x) = \int - 2 x^{- 3} + 20 x^{- 5} d x$

Step 4

Split the single integral into multiple integrals.

$\int - 2 x^{- 3} d x + \int 20 x^{- 5} d x$

Step 5

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$- 2 \int x^{- 3} d x + \int 20 x^{- 5} d x$

Step 6

By the Power Rule, the integral of $x^{- 3}$ with respect to $x$ is $- \frac{1}{2} x^{- 2}$ .

$- 2 (- \frac{1}{2} x^{- 2} + C) + \int 20 x^{- 5} d x$

Step 7

Simplify.

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Step 7.1

Combine $x^{- 2}$ and $\frac{1}{2}$ .

$- 2 (- \frac{x^{- 2}}{2} + C) + \int 20 x^{- 5} d x$

Step 7.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 2 (- \frac{1}{2 x^{2}} + C) + \int 20 x^{- 5} d x$

Step 8

Since $20$ is constant with respect to $x$ , move $20$ out of the integral.

$- 2 (- \frac{1}{2 x^{2}} + C) + 20 \int x^{- 5} d x$

Step 9

By the Power Rule, the integral of $x^{- 5}$ with respect to $x$ is $- \frac{1}{4} x^{- 4}$ .

$- 2 (- \frac{1}{2 x^{2}} + C) + 20 (- \frac{1}{4} x^{- 4} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

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Step 10.1.1

Combine $x^{- 4}$ and $\frac{1}{4}$ .

$- 2 (- \frac{1}{2 x^{2}} + C) + 20 (- \frac{x^{- 4}}{4} + C)$

Step 10.1.2

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 2 (- \frac{1}{2 x^{2}} + C) + 20 (- \frac{1}{4 x^{4}} + C)$

Step 10.2

Simplify.

$- \frac{- 1}{x^{2}} + 20 (- \frac{1}{4 x^{4}}) + C$

Step 10.3

Simplify.

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Step 10.3.1

Move the negative in front of the fraction.

$- - \frac{1}{x^{2}} + 20 (- \frac{1}{4 x^{4}}) + C$

Step 10.3.2

Multiply $- 1$ by $- 1$ .

$1 \frac{1}{x^{2}} + 20 (- \frac{1}{4 x^{4}}) + C$

Step 10.3.3

Multiply $\frac{1}{x^{2}}$ by $1$ .

$\frac{1}{x^{2}} + 20 (- \frac{1}{4 x^{4}}) + C$

Step 10.3.4

Multiply $- 1$ by $20$ .

$\frac{1}{x^{2}} - 20 \frac{1}{4 x^{4}} + C$

Step 10.3.5

Combine $- 20$ and $\frac{1}{4 x^{4}}$ .

$\frac{1}{x^{2}} + \frac{- 20}{4 x^{4}} + C$

Step 10.3.6

Cancel the common factor of $- 20$ and $4$ .

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Step 10.3.6.1

Factor $4$ out of $- 20$ .

$\frac{1}{x^{2}} + \frac{4 \cdot - 5}{4 x^{4}} + C$

Step 10.3.6.2

Cancel the common factors.

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Step 10.3.6.2.1

Factor $4$ out of $4 x^{4}$ .

$\frac{1}{x^{2}} + \frac{4 \cdot - 5}{4 (x^{4})} + C$

Step 10.3.6.2.2

Cancel the common factor.

$\frac{1}{x^{2}} + \frac{4 \cdot - 5}{4 x^{4}} + C$

Step 10.3.6.2.3

Rewrite the expression.

$\frac{1}{x^{2}} + \frac{- 5}{x^{4}} + C$

Step 10.3.7

Move the negative in front of the fraction.

$\frac{1}{x^{2}} - \frac{5}{x^{4}} + C$

Step 11

The answer is the antiderivative of the function $f (x) = - 2 x^{- 3} + 20 x^{- 5}$ .

$F (x) =$ $\frac{1}{x^{2}} - \frac{5}{x^{4}} + C$