Calculus Examples

$\int (\frac{\sqrt{x}}{2} + \frac{2}{\sqrt{x}}) d x$

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

$\frac{1}{2} (\frac{2}{3} x^{\frac{3}{2}} + C) + \int \frac{2}{\sqrt{x}} d x$

Step 6

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$\frac{1}{2} (\frac{2}{3} x^{\frac{3}{2}} + C) + 2 \int \frac{1}{\sqrt{x}} d x$

Step 7

Apply basic rules of exponents.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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Step 7.3.1

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

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

Step 7.3.2

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

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

Step 7.3.3

Move the negative in front of the fraction.

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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

Step 9.2

Simplify.

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Step 9.2.1

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

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

Step 9.2.2

Multiply $2$ by $3$ .

$\frac{2}{6} x^{\frac{3}{2}} + 2 \cdot 2 x^{\frac{1}{2}} + C$

Step 9.2.3

Cancel the common factor of $2$ and $6$ .

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Step 9.2.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{6} x^{\frac{3}{2}} + 2 \cdot 2 x^{\frac{1}{2}} + C$

Step 9.2.3.2

Cancel the common factors.

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Step 9.2.3.2.1

Factor $2$ out of $6$ .

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

Step 9.2.3.2.2

Cancel the common factor.

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

Step 9.2.3.2.3

Rewrite the expression.

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

Step 9.2.4

Multiply $2$ by $2$ .

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