Calculus Examples

$\int (\frac{4}{\sqrt{x}} - \frac{\sqrt{x}}{4}) d x$

Step 1

Remove parentheses.

$\int \frac{4}{\sqrt{x}} - \frac{\sqrt{x}}{4} d x$

Step 2

Split the single integral into multiple integrals.

$\int \frac{4}{\sqrt{x}} d x + \int - \frac{\sqrt{x}}{4} d x$

Step 3

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int \frac{1}{\sqrt{x}} d x + \int - \frac{\sqrt{x}}{4} d x$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

$4 \int {(x^{\frac{1}{2}})}^{- 1} d x + \int - \frac{\sqrt{x}}{4} d x$

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Move the negative in front of the fraction.

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

Step 5

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

$4 (2 x^{\frac{1}{2}} + C) + \int - \frac{\sqrt{x}}{4} d x$

Step 6

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$4 (2 x^{\frac{1}{2}} + C) - \int \frac{\sqrt{x}}{4} d x$

Step 7

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$4 (2 x^{\frac{1}{2}} + C) - (\frac{1}{4} \int \sqrt{x} d x)$

Step 8

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

$4 (2 x^{\frac{1}{2}} + C) - \frac{1}{4} \int x^{\frac{1}{2}} d x$

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

$8 x^{\frac{1}{2}} - \frac{1}{4} (\frac{2}{3} x^{\frac{3}{2}}) + C$

Step 10.2

Simplify.

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Step 10.2.1

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

$8 x^{\frac{1}{2}} - \frac{1}{4} \cdot \frac{2 x^{\frac{3}{2}}}{3} + C$

Step 10.2.2

Multiply $\frac{2 x^{\frac{3}{2}}}{3}$ by $\frac{1}{4}$ .

$8 x^{\frac{1}{2}} - \frac{2 x^{\frac{3}{2}}}{3 \cdot 4} + C$

Step 10.2.3

Multiply $3$ by $4$ .

$8 x^{\frac{1}{2}} - \frac{2 x^{\frac{3}{2}}}{12} + C$

Step 10.2.4

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

$8 x^{\frac{1}{2}} - \frac{2 (x^{\frac{3}{2}})}{12} + C$

Step 10.2.5

Cancel the common factors.

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Step 10.2.5.1

Factor $2$ out of $12$ .

$8 x^{\frac{1}{2}} - \frac{2 x^{\frac{3}{2}}}{2 \cdot 6} + C$

Step 10.2.5.2

Cancel the common factor.

$8 x^{\frac{1}{2}} - \frac{2 x^{\frac{3}{2}}}{2 \cdot 6} + C$

Step 10.2.5.3

Rewrite the expression.

$8 x^{\frac{1}{2}} - \frac{x^{\frac{3}{2}}}{6} + C$

Step 11

Reorder terms.

$8 x^{\frac{1}{2}} - \frac{1}{6} x^{\frac{3}{2}} + C$