Calculus Examples

$f (x) = \frac{1}{x^{2}} - \frac{2}{x^{4}}$

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.

$F (x) = \int \frac{1}{x^{2}} - \frac{2}{x^{4}} d x$

Step 3

Split the single integral into multiple integrals.

$\int \frac{1}{x^{2}} d x + \int - \frac{2}{x^{4}} d x$

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$\int {(x^{2})}^{- 1} d x + \int - \frac{2}{x^{4}} d x$

Step 4.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 4.2.1

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

$\int x^{2 \cdot - 1} d x + \int - \frac{2}{x^{4}} d x$

Step 4.2.2

Multiply $2$ by $- 1$ .

$\int x^{- 2} d x + \int - \frac{2}{x^{4}} d x$

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify the expression.

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Step 8.1

Multiply $2$ by $- 1$ .

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

Step 8.2

Move $x^{4}$ out of the denominator by raising it to the $- 1$ power.

$- x^{- 1} + C - 2 \int {(x^{4})}^{- 1} d x$

Step 8.3

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 8.3.1

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

$- x^{- 1} + C - 2 \int x^{4 \cdot - 1} d x$

Step 8.3.2

Multiply $4$ by $- 1$ .

$- x^{- 1} + C - 2 \int x^{- 4} d x$

Step 9

By the Power Rule, the integral of $x^{- 4}$ with respect to $x$ is $- \frac{1}{3} x^{- 3}$ .

$- x^{- 1} + C - 2 (- \frac{1}{3} x^{- 3} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

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Step 10.1.1

Combine $x^{- 3}$ and $\frac{1}{3}$ .

$- x^{- 1} + C - 2 (- \frac{x^{- 3}}{3} + C)$

Step 10.1.2

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- x^{- 1} + C - 2 (- \frac{1}{3 x^{3}} + C)$

Step 10.2

Simplify.

$- \frac{1}{x} - 2 (- \frac{1}{3 x^{3}}) + C$

Step 10.3

Simplify.

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Step 10.3.1

Multiply $- 1$ by $- 2$ .

$- \frac{1}{x} + 2 \frac{1}{3 x^{3}} + C$

Step 10.3.2

Combine $2$ and $\frac{1}{3 x^{3}}$ .

$- \frac{1}{x} + \frac{2}{3 x^{3}} + C$

Step 11

The answer is the antiderivative of the function $f (x) = \frac{1}{x^{2}} - \frac{2}{x^{4}}$ .

$F (x) =$ $- \frac{1}{x} + \frac{2}{3 x^{3}} + C$