Calculus Examples

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

Step 1

Simplify.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Move $x^{\frac{1}{2}}$ .

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

Step 1.2.2

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

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Step 1.2.2.1

Raise $x$ to the power of $1$ .

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

Step 1.2.2.2

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

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

Step 1.2.3

Write $1$ as a fraction with a common denominator.

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

Step 1.2.4

Combine the numerators over the common denominator.

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

Step 1.2.5

Add $1$ and $2$ .

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

Step 2

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

Multiply $3$ by $- 1$ .

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

Step 3.2.3

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

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

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

Step 5.2

Simplify.

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Step 5.2.1

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

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

Step 5.2.2

Move the negative in front of the fraction.

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

Step 5.2.3

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

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

Step 5.2.4

Cancel the common factor.

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

Step 5.2.5

Rewrite the expression.

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