Calculus Examples

$4 x^{- 2} + 6 \sqrt{x} - \frac{6}{x^{2}}$

Step 1

Write $4 x^{- 2} + 6 \sqrt{x} - \frac{6}{x^{2}}$ as a function.

$f (x) = 4 x^{- 2} + 6 \sqrt{x} - \frac{6}{x^{2}}$

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.

$F (x) = \int 4 x^{- 2} + 6 \sqrt{x} - \frac{6}{x^{2}} d x$

Step 4

Split the single integral into multiple integrals.

$\int 4 x^{- 2} d x + \int 6 \sqrt{x} d x + \int - \frac{6}{x^{2}} d x$

Step 5

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

$4 \int x^{- 2} d x + \int 6 \sqrt{x} d x + \int - \frac{6}{x^{2}} d x$

Step 6

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

$4 (- x^{- 1} + C) + \int 6 \sqrt{x} d x + \int - \frac{6}{x^{2}} d x$

Step 7

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$4 (- x^{- 1} + C) + 6 \int \sqrt{x} d x + \int - \frac{6}{x^{2}} d x$

Step 8

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$4 (- x^{- 1} + C) + 6 \int x^{\frac{1}{2}} d x + \int - \frac{6}{x^{2}} d x$

Step 9

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

$4 (- x^{- 1} + C) + 6 (\frac{2}{3} x^{\frac{3}{2}} + C) + \int - \frac{6}{x^{2}} d x$

Step 10

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

$4 (- x^{- 1} + C) + 6 (\frac{2}{3} x^{\frac{3}{2}} + C) - \int \frac{6}{x^{2}} d x$

Step 11

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$4 (- x^{- 1} + C) + 6 (\frac{2}{3} x^{\frac{3}{2}} + C) - (6 \int \frac{1}{x^{2}} d x)$

Step 12

Simplify the expression.

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Step 12.1

Simplify.

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Step 12.1.1

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

$4 (- x^{- 1} + C) + 6 (\frac{2 x^{\frac{3}{2}}}{3} + C) - (6 \int \frac{1}{x^{2}} d x)$

Step 12.1.2

Multiply $6$ by $- 1$ .

$4 (- x^{- 1} + C) + 6 (\frac{2 x^{\frac{3}{2}}}{3} + C) - 6 \int \frac{1}{x^{2}} d x$

Step 12.2

Apply basic rules of exponents.

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Step 12.2.1

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

$4 (- x^{- 1} + C) + 6 (\frac{2 x^{\frac{3}{2}}}{3} + C) - 6 \int {(x^{2})}^{- 1} d x$

Step 12.2.2

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

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Step 12.2.2.1

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

$4 (- x^{- 1} + C) + 6 (\frac{2 x^{\frac{3}{2}}}{3} + C) - 6 \int x^{2 \cdot - 1} d x$

Step 12.2.2.2

Multiply $2$ by $- 1$ .

$4 (- x^{- 1} + C) + 6 (\frac{2 x^{\frac{3}{2}}}{3} + C) - 6 \int x^{- 2} d x$

Step 13

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

$4 (- x^{- 1} + C) + 6 (\frac{2 x^{\frac{3}{2}}}{3} + C) - 6 (- x^{- 1} + C)$

Step 14

Simplify.

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Step 14.1

Simplify.

$- \frac{4}{x} + 4 x^{\frac{3}{2}} - 6 (- x^{- 1}) + C$

Step 14.2

Multiply $- 1$ by $- 6$ .

$- \frac{4}{x} + 4 x^{\frac{3}{2}} + 6 x^{- 1} + C$

Step 15

The answer is the antiderivative of the function $f (x) = 4 x^{- 2} + 6 \sqrt{x} - \frac{6}{x^{2}}$ .

$F (x) =$ $- \frac{4}{x} + 4 x^{\frac{3}{2}} + 6 x^{- 1} + C$