Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

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

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

Step 3

Apply basic rules of exponents.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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Step 3.3.2.1

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

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

Step 3.3.2.2

Multiply $3$ by $- 1$ .

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

Step 3.3.3

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

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

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

Step 5.2

Simplify.

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Step 5.2.1

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

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

Step 5.2.2

Move the negative in front of the fraction.

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

Step 5.2.3

Multiply $- 1$ by $2$ .

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

Step 5.2.4

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

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

Step 5.2.5

Multiply $- 2$ by $2$ .

$\frac{- 4}{x^{\frac{1}{2}}} + C$

Step 5.2.6

Move the negative in front of the fraction.

$- \frac{4}{x^{\frac{1}{2}}} + C$