Calculus Examples

$\frac{x}{\sqrt{x}}$

Step 1

Simplify the expression.

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Step 1.1

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

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

Step 1.2

Simplify.

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Step 1.2.1

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

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

Step 1.2.2

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

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Step 1.2.2.1

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

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Step 1.2.2.1.1

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

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

Step 1.2.2.1.2

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

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

Step 1.2.2.2

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

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

Step 1.2.2.3

Combine the numerators over the common denominator.

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

Step 1.2.2.4

Subtract $1$ from $2$ .

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

Step 2

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

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