Calculus Examples

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

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.

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 8

Simplify the expression.

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Step 8.1

Multiply $2$ by $- 1$ .

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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Step 8.4.1

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

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

Step 8.4.2

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

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

Step 8.4.3

Move the negative in front of the fraction.

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

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

Step 10.2

Multiply $- 2$ by $2$ .

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

Step 11

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

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