Calculus Examples

$\sqrt{x} - \frac{1}{x^{2}}$

Step 1

Write $\sqrt{x} - \frac{1}{x^{2}}$ as a function.

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

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.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Apply basic rules of exponents.

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Step 8.1

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

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

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

Multiply $2$ by $- 1$ .

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

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

Step 10.2

Simplify.

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Step 10.2.1

Multiply $- 1$ by $- 1$ .

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

Step 10.2.2

Multiply $x^{- 1}$ by $1$ .

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

Step 11

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

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