Calculus Examples

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

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.

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 5

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

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

Step 6

Simplify the expression.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

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

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

Step 6.2.2

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

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Step 6.2.2.1

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

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

Step 6.2.2.2

Multiply $2$ by $- 1$ .

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

Step 7

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

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

Step 8

Simplify.

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

Step 9

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

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