Calculus Examples

$f (x) = \frac{1}{2} x^{\frac{1}{3}} - \frac{2}{x^{3}} + \frac{3}{\sqrt{x}}$

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.

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 5

By the Power Rule, the integral of $x^{\frac{1}{3}}$ with respect to $x$ is $\frac{3}{4} x^{\frac{4}{3}}$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify the expression.

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Step 8.1

Multiply $2$ by $- 1$ .

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

Step 8.2

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

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

Step 8.3

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 8.3.1

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

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

Step 8.3.2

Multiply $3$ by $- 1$ .

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

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

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

Step 10.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 11

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 12

Apply basic rules of exponents.

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

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

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Step 12.3.1

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

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

Step 12.3.2

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

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

Step 12.3.3

Move the negative in front of the fraction.

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

Step 13

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

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

Step 14

Simplify.

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Step 14.1

Simplify.

$\frac{3 x^{\frac{4}{3}}}{8} + \frac{1}{x^{2}} + 3 (2 x^{\frac{1}{2}}) + C$

Step 14.2

Simplify the expression.

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Step 14.2.1

Multiply $2$ by $3$ .

$\frac{3 x^{\frac{4}{3}}}{8} + \frac{1}{x^{2}} + 6 x^{\frac{1}{2}} + C$

Step 14.2.2

Reorder terms.

$\frac{3}{8} x^{\frac{4}{3}} + \frac{1}{x^{2}} + 6 x^{\frac{1}{2}} + C$

Step 15

The answer is the antiderivative of the function $f (x) = \frac{1}{2} x^{\frac{1}{3}} - \frac{2}{x^{3}} + \frac{3}{\sqrt{x}}$ .

$F (x) =$ $\frac{3}{8} x^{\frac{4}{3}} + \frac{1}{x^{2}} + 6 x^{\frac{1}{2}} + C$