Calculus Examples

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

Step 1

Simplify the expression.

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Step 1.1

Simplify.

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 1.1.2

Multiply $x^{2}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 1.1.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.1.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.1.2.3

Combine $2$ and $\frac{2}{2}$ .

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

Step 1.1.2.4

Combine the numerators over the common denominator.

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

Step 1.1.2.5

Simplify the numerator.

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Step 1.1.2.5.1

Multiply $2$ by $2$ .

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

Step 1.1.2.5.2

Add $4$ and $1$ .

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

Step 1.2

Apply basic rules of exponents.

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Step 1.2.1

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

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

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

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Step 1.2.2.2.1

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

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

Step 1.2.2.2.2

Multiply $5$ by $- 1$ .

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

Step 1.2.2.3

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

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

Step 3.2

Simplify.

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Step 3.2.1

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

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

Step 3.2.2

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

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