Calculus Examples

$\frac{1}{\sqrt{x}} - \frac{2}{\sqrt[3]{x}}$

Step 1

Write $\frac{1}{\sqrt{x}} - \frac{2}{\sqrt[3]{x}}$ as a function.

$f (x) = \frac{1}{\sqrt{x}} - \frac{2}{\sqrt[3]{x}}$

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.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Apply basic rules of exponents.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify the expression.

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Step 9.1

Multiply $2$ by $- 1$ .

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

Step 9.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

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

Step 9.3

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

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

Step 9.4

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

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Step 9.4.1

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

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

Step 9.4.2

Combine $\frac{1}{3}$ and $- 1$ .

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

Step 9.4.3

Move the negative in front of the fraction.

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

Simplify.

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

Step 11.2

Simplify.

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Step 11.2.1

Combine $- 2$ and $\frac{3}{2}$ .

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

Step 11.2.2

Multiply $- 2$ by $3$ .

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

Step 11.2.3

Cancel the common factor of $- 6$ and $2$ .

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Step 11.2.3.1

Factor $2$ out of $- 6$ .

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

Step 11.2.3.2

Cancel the common factors.

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Step 11.2.3.2.1

Factor $2$ out of $2$ .

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

Step 11.2.3.2.2

Cancel the common factor.

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

Step 11.2.3.2.3

Rewrite the expression.

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

Step 11.2.3.2.4

Divide $- 3$ by $1$ .

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

Step 12

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

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