Calculus Examples

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

Step 1

Apply basic rules of exponents.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

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

Tap for more steps...

Step 1.2.2.1

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

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

Step 1.2.2.2

Multiply $3$ by $- 1$ .

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

Step 1.2.3

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Rewrite $- 2 x^{- \frac{1}{2}} + C$ as $- 2 \frac{1}{x^{\frac{1}{2}}} + C$ .

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

Step 3.2

Simplify.

Tap for more steps...

Step 3.2.1

Combine $- 2$ and $\frac{1}{x^{\frac{1}{2}}}$ .

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

Step 3.2.2

Move the negative in front of the fraction.

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