Calculus Examples

$\int \frac{3}{4 \sqrt{x^{3}}} d x$

Step 1

Since $\frac{3}{4}$ is constant with respect to $x$ , move $\frac{3}{4}$ out of the integral.

$\frac{3}{4} \int \frac{1}{\sqrt{x^{3}}} d x$

Step 2

Apply basic rules of exponents.

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Step 2.1

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

$\frac{3}{4} \int \frac{1}{x^{\frac{3}{2}}} d x$

Step 2.2

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

$\frac{3}{4} \int {(x^{\frac{3}{2}})}^{- 1} d x$

Step 2.3

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

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Step 2.3.1

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

$\frac{3}{4} \int x^{\frac{3}{2} \cdot - 1} d x$

Step 2.3.2

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

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Step 2.3.2.1

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

$\frac{3}{4} \int x^{\frac{3 \cdot - 1}{2}} d x$

Step 2.3.2.2

Multiply $3$ by $- 1$ .

$\frac{3}{4} \int x^{\frac{- 3}{2}} d x$

Step 2.3.3

Move the negative in front of the fraction.

$\frac{3}{4} \int x^{- \frac{3}{2}} d x$

Step 3

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

$\frac{3}{4} (- 2 x^{- \frac{1}{2}} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $\frac{3}{4} (- 2 x^{- \frac{1}{2}} + C)$ as $\frac{3}{4} (- 2 \frac{1}{x^{\frac{1}{2}}}) + C$ .

$\frac{3}{4} (- 2 \frac{1}{x^{\frac{1}{2}}}) + C$

Step 4.2

Simplify.

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Step 4.2.1

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

$\frac{3}{4} \cdot \frac{- 2}{x^{\frac{1}{2}}} + C$

Step 4.2.2

Move the negative in front of the fraction.

$\frac{3}{4} (- \frac{2}{x^{\frac{1}{2}}}) + C$

Step 4.2.3

Multiply $\frac{3}{4}$ by $\frac{2}{x^{\frac{1}{2}}}$ .

$- \frac{3 \cdot 2}{4 x^{\frac{1}{2}}} + C$

Step 4.2.4

Multiply $3$ by $2$ .

$- \frac{6}{4 x^{\frac{1}{2}}} + C$

Step 4.2.5

Factor $2$ out of $6$ .

$- \frac{2 (3)}{4 x^{\frac{1}{2}}} + C$

Step 4.2.6

Cancel the common factors.

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Step 4.2.6.1

Factor $2$ out of $4 x^{\frac{1}{2}}$ .

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

Step 4.2.6.2

Cancel the common factor.

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

Step 4.2.6.3

Rewrite the expression.

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