Calculus Examples

$\frac{1}{7 \sqrt[7]{x^{6}}}$

Step 1

Since $\frac{1}{7}$ is constant with respect to $x$ , move $\frac{1}{7}$ out of the integral.

$\frac{1}{7} \int \frac{1}{\sqrt[7]{x^{6}}} d x$

Step 2

Apply basic rules of exponents.

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{x^{6}}$ as $x^{\frac{6}{7}}$ .

$\frac{1}{7} \int \frac{1}{x^{\frac{6}{7}}} d x$

Step 2.2

Move $x^{\frac{6}{7}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{7} \int {(x^{\frac{6}{7}})}^{- 1} d x$

Step 2.3

Multiply the exponents in ${(x^{\frac{6}{7}})}^{- 1}$ .

Tap for more steps...

Step 2.3.1

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

$\frac{1}{7} \int x^{\frac{6}{7} \cdot - 1} d x$

Step 2.3.2

Multiply $\frac{6}{7} \cdot - 1$ .

Tap for more steps...

Step 2.3.2.1

Combine $\frac{6}{7}$ and $- 1$ .

$\frac{1}{7} \int x^{\frac{6 \cdot - 1}{7}} d x$

Step 2.3.2.2

Multiply $6$ by $- 1$ .

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

Step 2.3.3

Move the negative in front of the fraction.

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

Step 3

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

$\frac{1}{7} (7 x^{\frac{1}{7}} + C)$

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

Rewrite $\frac{1}{7} (7 x^{\frac{1}{7}} + C)$ as $\frac{1}{7} \cdot 7 x^{\frac{1}{7}} + C$ .

$\frac{1}{7} \cdot 7 x^{\frac{1}{7}} + C$

Step 4.2

Simplify.

Tap for more steps...

Step 4.2.1

Combine $\frac{1}{7}$ and $7$ .

$\frac{7}{7} x^{\frac{1}{7}} + C$

Step 4.2.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 4.2.2.1

Cancel the common factor.

$\frac{7}{7} x^{\frac{1}{7}} + C$

Step 4.2.2.2

Rewrite the expression.

$1 x^{\frac{1}{7}} + C$

Step 4.2.3

Multiply $x^{\frac{1}{7}}$ by $1$ .

$x^{\frac{1}{7}} + C$