Calculus Examples

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

Step 1

Simplify.

Tap for more steps...

Step 1.1

Simplify the numerator.

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

Factor $x^{\frac{1}{3}}$ out of $x^{\frac{1}{2}} + x^{\frac{1}{3}}$ .

Tap for more steps...

Step 1.1.3.1

Factor $x^{\frac{1}{3}}$ out of $x^{\frac{1}{2}}$ .

$\int \frac{x^{\frac{1}{3}} x^{\frac{1}{6}} + x^{\frac{1}{3}}}{x} d x$

Step 1.1.3.2

Multiply by $1$ .

$\int \frac{x^{\frac{1}{3}} x^{\frac{1}{6}} + x^{\frac{1}{3}} \cdot 1}{x} d x$

Step 1.1.3.3

Factor $x^{\frac{1}{3}}$ out of $x^{\frac{1}{3}} x^{\frac{1}{6}} + x^{\frac{1}{3}} \cdot 1$ .

$\int \frac{x^{\frac{1}{3}} (x^{\frac{1}{6}} + 1)}{x} d x$

Step 1.2

Move $x^{\frac{1}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\int \frac{x^{\frac{1}{6}} + 1}{x \cdot x^{- \frac{1}{3}}} d x$

Step 1.3

Multiply $x$ by $x^{- \frac{1}{3}}$ by adding the exponents.

Tap for more steps...

Step 1.3.1

Multiply $x$ by $x^{- \frac{1}{3}}$ .

Tap for more steps...

Step 1.3.1.1

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

$\int \frac{x^{\frac{1}{6}} + 1}{x^{1} x^{- \frac{1}{3}}} d x$

Step 1.3.1.2

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

$\int \frac{x^{\frac{1}{6}} + 1}{x^{1 - \frac{1}{3}}} d x$

Step 1.3.2

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

$\int \frac{x^{\frac{1}{6}} + 1}{x^{\frac{3}{3} - \frac{1}{3}}} d x$

Step 1.3.3

Combine the numerators over the common denominator.

$\int \frac{x^{\frac{1}{6}} + 1}{x^{\frac{3 - 1}{3}}} d x$

Step 1.3.4

Subtract $1$ from $3$ .

$\int \frac{x^{\frac{1}{6}} + 1}{x^{\frac{2}{3}}} d x$

Step 2

Move $x^{\frac{2}{3}}$ out of the denominator by raising it to the $- 1$ power.

$\int (x^{\frac{1}{6}} + 1) {(x^{\frac{2}{3}})}^{- 1} d x$

Step 3

Multiply the exponents in ${(x^{\frac{2}{3}})}^{- 1}$ .

Tap for more steps...

Step 3.1

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

$\int (x^{\frac{1}{6}} + 1) x^{\frac{2}{3} \cdot - 1} d x$

Step 3.2

Multiply $\frac{2}{3} \cdot - 1$ .

Tap for more steps...

Step 3.2.1

Combine $\frac{2}{3}$ and $- 1$ .

$\int (x^{\frac{1}{6}} + 1) x^{\frac{2 \cdot - 1}{3}} d x$

Step 3.2.2

Multiply $2$ by $- 1$ .

$\int (x^{\frac{1}{6}} + 1) x^{\frac{- 2}{3}} d x$

Step 3.3

Move the negative in front of the fraction.

$\int (x^{\frac{1}{6}} + 1) x^{- \frac{2}{3}} d x$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

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

Step 4.2

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

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

Step 4.3

To write $- \frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\int x^{\frac{1}{6} - \frac{2}{3} \cdot \frac{2}{2}} + 1 x^{- \frac{2}{3}} d x$

Step 4.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.4.1

Multiply $\frac{2}{3}$ by $\frac{2}{2}$ .

$\int x^{\frac{1}{6} - \frac{2 \cdot 2}{3 \cdot 2}} + 1 x^{- \frac{2}{3}} d x$

Step 4.4.2

Multiply $3$ by $2$ .

$\int x^{\frac{1}{6} - \frac{2 \cdot 2}{6}} + 1 x^{- \frac{2}{3}} d x$

Step 4.5

Combine the numerators over the common denominator.

$\int x^{\frac{1 - 2 \cdot 2}{6}} + 1 x^{- \frac{2}{3}} d x$

Step 4.6

Simplify the numerator.

Tap for more steps...

Step 4.6.1

Multiply $- 2$ by $2$ .

$\int x^{\frac{1 - 4}{6}} + 1 x^{- \frac{2}{3}} d x$

Step 4.6.2

Subtract $4$ from $1$ .

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

Step 4.7

Cancel the common factor of $- 3$ and $6$ .

Tap for more steps...

Step 4.7.1

Factor $3$ out of $- 3$ .

$\int x^{\frac{3 (- 1)}{6}} + 1 x^{- \frac{2}{3}} d x$

Step 4.7.2

Cancel the common factors.

Tap for more steps...

Step 4.7.2.1

Factor $3$ out of $6$ .

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

Step 4.7.2.2

Cancel the common factor.

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

Step 4.7.2.3

Rewrite the expression.

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

Step 4.8

Multiply $x^{- \frac{2}{3}}$ by $1$ .

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

Step 5

Move the negative in front of the fraction.

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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