Calculus Examples

$\int (\frac{3}{\sqrt[5]{x^{2}}} - \frac{2}{\sqrt[5]{x}}) d x$

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 4

Apply basic rules of exponents.

Tap for more steps...

Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{2}}$ as $x^{\frac{2}{5}}$ .

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

Step 4.2

Move $x^{\frac{2}{5}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 4.3

Multiply the exponents in ${(x^{\frac{2}{5}})}^{- 1}$ .

Tap for more steps...

Step 4.3.1

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

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

Step 4.3.2

Multiply $\frac{2}{5} \cdot - 1$ .

Tap for more steps...

Step 4.3.2.1

Combine $\frac{2}{5}$ and $- 1$ .

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

Step 4.3.2.2

Multiply $2$ by $- 1$ .

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

Step 4.3.3

Move the negative in front of the fraction.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify the expression.

Tap for more steps...

Step 8.1

Simplify.

Tap for more steps...

Step 8.1.1

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

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

Step 8.1.2

Multiply $2$ by $- 1$ .

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

Step 8.2

Apply basic rules of exponents.

Tap for more steps...

Step 8.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

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

Step 8.2.2

Move $x^{\frac{1}{5}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 8.2.3

Multiply the exponents in ${(x^{\frac{1}{5}})}^{- 1}$ .

Tap for more steps...

Step 8.2.3.1

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

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

Step 8.2.3.2

Combine $\frac{1}{5}$ and $- 1$ .

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

Step 8.2.3.3

Move the negative in front of the fraction.

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

Step 9

By the Power Rule, the integral of $x^{- \frac{1}{5}}$ with respect to $x$ is $\frac{5}{4} x^{\frac{4}{5}}$ .

$3 (\frac{5 x^{\frac{3}{5}}}{3} + C) - 2 (\frac{5}{4} x^{\frac{4}{5}} + C)$

Step 10

Simplify.

Tap for more steps...

Step 10.1

Combine $\frac{5}{4}$ and $x^{\frac{4}{5}}$ .

$3 (\frac{5 x^{\frac{3}{5}}}{3} + C) - 2 (\frac{5 x^{\frac{4}{5}}}{4} + C)$

Step 10.2

Simplify.

$5 x^{\frac{3}{5}} - 2 \frac{5 x^{\frac{4}{5}}}{4} + C$

Step 10.3

Simplify.

Tap for more steps...

Step 10.3.1

Combine $- 2$ and $\frac{5 x^{\frac{4}{5}}}{4}$ .

$5 x^{\frac{3}{5}} + \frac{- 2 (5 x^{\frac{4}{5}})}{4} + C$

Step 10.3.2

Multiply $5$ by $- 2$ .

$5 x^{\frac{3}{5}} + \frac{- 10 x^{\frac{4}{5}}}{4} + C$

Step 10.3.3

Factor $2$ out of $- 10 x^{\frac{4}{5}}$ .

$5 x^{\frac{3}{5}} + \frac{2 (- 5 x^{\frac{4}{5}})}{4} + C$

Step 10.3.4

Cancel the common factors.

Tap for more steps...

Step 10.3.4.1

Factor $2$ out of $4$ .

$5 x^{\frac{3}{5}} + \frac{2 (- 5 x^{\frac{4}{5}})}{2 (2)} + C$

Step 10.3.4.2

Cancel the common factor.

$5 x^{\frac{3}{5}} + \frac{2 (- 5 x^{\frac{4}{5}})}{2 \cdot 2} + C$

Step 10.3.4.3

Rewrite the expression.

$5 x^{\frac{3}{5}} + \frac{- 5 x^{\frac{4}{5}}}{2} + C$

Step 10.3.5

Move the negative in front of the fraction.

$5 x^{\frac{3}{5}} - \frac{5 x^{\frac{4}{5}}}{2} + C$

Step 11

Reorder terms.

$5 x^{\frac{3}{5}} - \frac{5}{2} x^{\frac{4}{5}} + C$