Calculus Examples

$\int \frac{3}{\sqrt{x^{5}}} d x$

Step 1

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

$3 \int \frac{1}{\sqrt{x^{5}}} d x$

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 $\sqrt{x^{5}}$ as $x^{\frac{5}{2}}$ .

$3 \int \frac{1}{x^{\frac{5}{2}}} d x$

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

$3 \int x^{\frac{5}{2} \cdot - 1} d x$

Step 2.3.2

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

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Step 2.3.2.1

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

$3 \int x^{\frac{5 \cdot - 1}{2}} d x$

Step 2.3.2.2

Multiply $5$ by $- 1$ .

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

Step 2.3.3

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

Simplify.

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Step 4.1.1

Combine $x^{- \frac{3}{2}}$ and $\frac{2}{3}$ .

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

Step 4.1.2

Move $2$ to the left of $x^{- \frac{3}{2}}$ .

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

Step 4.1.3

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.2

Simplify.

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

Step 4.3

Simplify.

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Step 4.3.1

Multiply $- 1$ by $3$ .

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

Step 4.3.2

Combine $- 3$ and $\frac{2}{3 x^{\frac{3}{2}}}$ .

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

Step 4.3.3

Multiply $- 3$ by $2$ .

$\frac{- 6}{3 x^{\frac{3}{2}}} + C$

Step 4.3.4

Factor $3$ out of $- 6$ .

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

Step 4.3.5

Cancel the common factors.

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Step 4.3.5.1

Factor $3$ out of $3 x^{\frac{3}{2}}$ .

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

Step 4.3.5.2

Cancel the common factor.

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

Step 4.3.5.3

Rewrite the expression.

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

Step 4.3.6

Move the negative in front of the fraction.

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