Calculus Examples

$g (x) = 5 \sqrt[3]{x^{2}} - \frac{1}{2 \sqrt{x}} + π$

Step 1

The function $G (x)$ can be found by finding the indefinite integral of the derivative $g (x)$ .

$G (x) = \int g (x) d x$

Step 2

Set up the integral to solve.

$G (x) = \int 5 \sqrt[3]{x^{2}} - \frac{1}{2 \sqrt{x}} + π d x$

Step 3

Split the single integral into multiple integrals.

$\int 5 \sqrt[3]{x^{2}} d x + \int - \frac{1}{2 \sqrt{x}} d x + \int π d x$

Step 4

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int \sqrt[3]{x^{2}} d x + \int - \frac{1}{2 \sqrt{x}} d x + \int π d x$

Step 5

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$5 \int x^{\frac{2}{3}} d x + \int - \frac{1}{2 \sqrt{x}} d x + \int π d x$

Step 6

By the Power Rule, the integral of $x^{\frac{2}{3}}$ with respect to $x$ is $\frac{3}{5} x^{\frac{5}{3}}$ .

$5 (\frac{3}{5} x^{\frac{5}{3}} + C) + \int - \frac{1}{2 \sqrt{x}} d x + \int π d x$

Step 7

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

$5 (\frac{3}{5} x^{\frac{5}{3}} + C) - \int \frac{1}{2 \sqrt{x}} d x + \int π d x$

Step 8

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

$5 (\frac{3}{5} x^{\frac{5}{3}} + C) - (\frac{1}{2} \int \frac{1}{\sqrt{x}} d x) + \int π d x$

Step 9

Simplify the expression.

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Step 9.1

Combine $\frac{3}{5}$ and $x^{\frac{5}{3}}$ .

$5 (\frac{3 x^{\frac{5}{3}}}{5} + C) - \frac{1}{2} \int \frac{1}{\sqrt{x}} d x + \int π d x$

Step 9.2

Apply basic rules of exponents.

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Step 9.2.1

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

$5 (\frac{3 x^{\frac{5}{3}}}{5} + C) - \frac{1}{2} \int \frac{1}{x^{\frac{1}{2}}} d x + \int π d x$

Step 9.2.2

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

$5 (\frac{3 x^{\frac{5}{3}}}{5} + C) - \frac{1}{2} \int {(x^{\frac{1}{2}})}^{- 1} d x + \int π d x$

Step 9.2.3

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

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Step 9.2.3.1

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

$5 (\frac{3 x^{\frac{5}{3}}}{5} + C) - \frac{1}{2} \int x^{\frac{1}{2} \cdot - 1} d x + \int π d x$

Step 9.2.3.2

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

$5 (\frac{3 x^{\frac{5}{3}}}{5} + C) - \frac{1}{2} \int x^{\frac{- 1}{2}} d x + \int π d x$

Step 9.2.3.3

Move the negative in front of the fraction.

$5 (\frac{3 x^{\frac{5}{3}}}{5} + C) - \frac{1}{2} \int x^{- \frac{1}{2}} d x + \int π d x$

Step 10

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

$5 (\frac{3 x^{\frac{5}{3}}}{5} + C) - \frac{1}{2} (2 x^{\frac{1}{2}} + C) + \int π d x$

Step 11

Apply the constant rule.

$5 (\frac{3 x^{\frac{5}{3}}}{5} + C) - \frac{1}{2} (2 x^{\frac{1}{2}} + C) + π x + C$

Step 12

Simplify.

$3 x^{\frac{5}{3}} - x^{\frac{1}{2}} + π x + C$

Step 13

The answer is the antiderivative of the function $g (x) = 5 \sqrt[3]{x^{2}} - \frac{1}{2 \sqrt{x}} + π$ .

$G (x) =$ $3 x^{\frac{5}{3}} - x^{\frac{1}{2}} + π x + C$