Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Move the negative in front of the fraction.

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Apply the distributive property.

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

Step 5.2

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

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

Step 5.3

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 5.4

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.5

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

Tap for more steps...

Step 5.5.1

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

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

Step 5.5.2

Multiply $2$ by $3$ .

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

Step 5.5.3

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

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

Step 5.5.4

Multiply $3$ by $2$ .

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

Step 5.6

Combine the numerators over the common denominator.

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

Step 5.7

Subtract $2$ from $3$ .

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

Step 5.8

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

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

By the Power Rule, the integral of $x^{\frac{1}{6}}$ with respect to $x$ is $\frac{6}{7} x^{\frac{7}{6}}$ .

$\frac{6}{7} x^{\frac{7}{6}} + C + \int x^{- \frac{1}{3}} d x$

Step 8

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

$\frac{6}{7} x^{\frac{7}{6}} + C + \frac{3}{2} x^{\frac{2}{3}} + C$

Step 9

Simplify.

$\frac{6}{7} x^{\frac{7}{6}} + \frac{3}{2} x^{\frac{2}{3}} + C$