Calculus Examples

$f (x) = 8 x^{4} + \frac{x^{- 2}}{2}$

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.

$F (x) = \int 8 x^{4} + \frac{x^{- 2}}{2} d x$

Step 3

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

$\int 8 x^{4} + \frac{1}{2 x^{2}} d x$

Step 4

Split the single integral into multiple integrals.

$\int 8 x^{4} d x + \int \frac{1}{2 x^{2}} d x$

Step 5

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

$8 \int x^{4} d x + \int \frac{1}{2 x^{2}} d x$

Step 6

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

$8 (\frac{1}{5} x^{5} + C) + \int \frac{1}{2 x^{2}} d x$

Step 7

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$8 (\frac{1}{5} x^{5} + C) + \frac{1}{2} \int \frac{1}{x^{2}} d x$

Step 8

Simplify the expression.

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Step 8.1

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

$8 (\frac{1}{5} x^{5} + C) + \frac{1}{2} \int {(x^{2})}^{- 1} d x$

Step 8.2

Simplify.

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Step 8.2.1

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

$8 (\frac{x^{5}}{5} + C) + \frac{1}{2} \int {(x^{2})}^{- 1} d x$

Step 8.2.2

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

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Step 8.2.2.1

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

$8 (\frac{x^{5}}{5} + C) + \frac{1}{2} \int x^{2 \cdot - 1} d x$

Step 8.2.2.2

Multiply $2$ by $- 1$ .

$8 (\frac{x^{5}}{5} + C) + \frac{1}{2} \int x^{- 2} d x$

Step 9

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

$8 (\frac{x^{5}}{5} + C) + \frac{1}{2} (- x^{- 1} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$\frac{8 x^{5}}{5} + \frac{1}{2} (- x^{- 1}) + C$

Step 10.2

Simplify.

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Step 10.2.1

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

$\frac{8 x^{5}}{5} - \frac{x^{- 1}}{2} + C$

Step 10.2.2

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

$\frac{8 x^{5}}{5} - \frac{1}{2 x} + C$

Step 10.3

Reorder terms.

$\frac{8}{5} x^{5} - \frac{1}{2 x} + C$

Step 11

The answer is the antiderivative of the function $f (x) = 8 x^{4} + \frac{x^{- 2}}{2}$ .

$F (x) =$ $\frac{8}{5} x^{5} - \frac{1}{2 x} + C$