Calculus Examples

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

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.

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

Apply basic rules of exponents.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Multiply $2$ by $- 1$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Simplify.

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Step 8.2.1

Multiply $- 1$ by $- 1$ .

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

Step 8.2.2

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

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

Step 9

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

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