Calculus Examples

$\frac{5 x + 2}{\sqrt[3]{x}}$

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Move the negative in front of the fraction.

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

Step 4

Expand $(5 x + 2) x^{- \frac{1}{3}}$ .

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Step 4.1

Apply the distributive property.

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

Step 4.2

Raise $x$ to the power of $1$ .

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

Step 4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.4

Write $1$ as a fraction with a common denominator.

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Subtract $1$ from $3$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

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

Step 10.2

Simplify.

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Step 10.2.1

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

$\frac{5 \cdot 3}{5} x^{\frac{5}{3}} + 2 (\frac{3}{2}) x^{\frac{2}{3}} + C$

Step 10.2.2

Multiply $5$ by $3$ .

$\frac{15}{5} x^{\frac{5}{3}} + 2 (\frac{3}{2}) x^{\frac{2}{3}} + C$

Step 10.2.3

Cancel the common factor of $15$ and $5$ .

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Step 10.2.3.1

Factor $5$ out of $15$ .

$\frac{5 \cdot 3}{5} x^{\frac{5}{3}} + 2 (\frac{3}{2}) x^{\frac{2}{3}} + C$

Step 10.2.3.2

Cancel the common factors.

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Step 10.2.3.2.1

Factor $5$ out of $5$ .

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

Step 10.2.3.2.2

Cancel the common factor.

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

Step 10.2.3.2.3

Rewrite the expression.

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

Step 10.2.3.2.4

Divide $3$ by $1$ .

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

Step 10.2.4

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

$3 x^{\frac{5}{3}} + \frac{2 \cdot 3}{2} x^{\frac{2}{3}} + C$

Step 10.2.5

Multiply $2$ by $3$ .

$3 x^{\frac{5}{3}} + \frac{6}{2} x^{\frac{2}{3}} + C$

Step 10.2.6

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

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Step 10.2.6.1

Factor $2$ out of $6$ .

$3 x^{\frac{5}{3}} + \frac{2 \cdot 3}{2} x^{\frac{2}{3}} + C$

Step 10.2.6.2

Cancel the common factors.

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Step 10.2.6.2.1

Factor $2$ out of $2$ .

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

Step 10.2.6.2.2

Cancel the common factor.

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

Step 10.2.6.2.3

Rewrite the expression.

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

Step 10.2.6.2.4

Divide $3$ by $1$ .

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