Calculus Examples

$\int \frac{\sqrt{5 x}}{5} + \frac{5}{\sqrt{5 x}} d x$

Step 1

Split the single integral into multiple integrals.

$\int \frac{\sqrt{5 x}}{5} d x + \int \frac{5}{\sqrt{5 x}} d x$

Step 2

Since $\frac{1}{5}$ is constant with respect to $x$ , move $\frac{1}{5}$ out of the integral.

$\frac{1}{5} \int \sqrt{5 x} d x + \int \frac{5}{\sqrt{5 x}} d x$

Step 3

Let $u_{1} = 5 x$ . Then $d u_{1} = 5 d x$ , so $\frac{1}{5} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 3.1

Let $u_{1} = 5 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 3.1.1

Differentiate $5 x$ .

$\frac{d}{d x} [5 x]$

Step 3.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x]$

Step 3.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$5 \cdot 1$

Step 3.1.4

Multiply $5$ by $1$ .

$5$

Step 3.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\frac{1}{5} \int \sqrt{u_{1}} \frac{1}{5} d u_{1} + \int \frac{5}{\sqrt{5 x}} d x$

Step 4

Combine $\sqrt{u_{1}}$ and $\frac{1}{5}$ .

$\frac{1}{5} \int \frac{\sqrt{u_{1}}}{5} d u_{1} + \int \frac{5}{\sqrt{5 x}} d x$

Step 5

Since $\frac{1}{5}$ is constant with respect to $u_{1}$ , move $\frac{1}{5}$ out of the integral.

$\frac{1}{5} (\frac{1}{5} \int \sqrt{u_{1}} d u_{1}) + \int \frac{5}{\sqrt{5 x}} d x$

Step 6

Simplify the expression.

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Step 6.1

Simplify.

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Step 6.1.1

Multiply $\frac{1}{5}$ by $\frac{1}{5}$ .

$\frac{1}{5 \cdot 5} \int \sqrt{u_{1}} d u_{1} + \int \frac{5}{\sqrt{5 x}} d x$

Step 6.1.2

Multiply $5$ by $5$ .

$\frac{1}{25} \int \sqrt{u_{1}} d u_{1} + \int \frac{5}{\sqrt{5 x}} d x$

Step 6.2

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

$\frac{1}{25} \int {u_{1}}^{\frac{1}{2}} d u_{1} + \int \frac{5}{\sqrt{5 x}} d x$

Step 7

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

$\frac{1}{25} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C) + \int \frac{5}{\sqrt{5 x}} d x$

Step 8

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$\frac{1}{25} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C) + 5 \int \frac{1}{\sqrt{5 x}} d x$

Step 9

Let $u_{2} = 5 x$ . Then $d u_{2} = 5 d x$ , so $\frac{1}{5} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 9.1

Let $u_{2} = 5 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 9.1.1

Differentiate $5 x$ .

$\frac{d}{d x} [5 x]$

Step 9.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x]$

Step 9.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$5 \cdot 1$

Step 9.1.4

Multiply $5$ by $1$ .

$5$

Step 9.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{25} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C) + 5 \int \frac{1}{\sqrt{u_{2}}} \cdot \frac{1}{5} d u_{2}$

Step 10

Simplify.

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Step 10.1

Multiply $\frac{1}{\sqrt{u_{2}}}$ by $\frac{1}{5}$ .

$\frac{1}{25} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C) + 5 \int \frac{1}{\sqrt{u_{2}} \cdot 5} d u_{2}$

Step 10.2

Move $5$ to the left of $\sqrt{u_{2}}$ .

$\frac{1}{25} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C) + 5 \int \frac{1}{5 \sqrt{u_{2}}} d u_{2}$

Step 11

Since $\frac{1}{5}$ is constant with respect to $u_{2}$ , move $\frac{1}{5}$ out of the integral.

$\frac{1}{25} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C) + 5 (\frac{1}{5} \int \frac{1}{\sqrt{u_{2}}} d u_{2})$

Step 12

Simplify the expression.

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Step 12.1

Simplify.

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Step 12.1.1

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

$\frac{1}{25} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C) + \frac{5}{5} \int \frac{1}{\sqrt{u_{2}}} d u_{2}$

Step 12.1.2

Cancel the common factor of $5$ .

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Step 12.1.2.1

Cancel the common factor.

$\frac{1}{25} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C) + \frac{5}{5} \int \frac{1}{\sqrt{u_{2}}} d u_{2}$

Step 12.1.2.2

Rewrite the expression.

$\frac{1}{25} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C) + 1 \int \frac{1}{\sqrt{u_{2}}} d u_{2}$

Step 12.1.3

Multiply $\int \frac{1}{\sqrt{u_{2}}} d u_{2}$ by $1$ .

$\frac{1}{25} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C) + \int \frac{1}{\sqrt{u_{2}}} d u_{2}$

Step 12.2

Apply basic rules of exponents.

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Step 12.2.1

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

$\frac{1}{25} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C) + \int \frac{1}{{u_{2}}^{\frac{1}{2}}} d u_{2}$

Step 12.2.2

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

$\frac{1}{25} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C) + \int {({u_{2}}^{\frac{1}{2}})}^{- 1} d u_{2}$

Step 12.2.3

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

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Step 12.2.3.1

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

$\frac{1}{25} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C) + \int {u_{2}}^{\frac{1}{2} \cdot - 1} d u_{2}$

Step 12.2.3.2

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

$\frac{1}{25} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C) + \int {u_{2}}^{\frac{- 1}{2}} d u_{2}$

Step 12.2.3.3

Move the negative in front of the fraction.

$\frac{1}{25} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C) + \int {u_{2}}^{- \frac{1}{2}} d u_{2}$

Step 13

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

$\frac{1}{25} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C) + 2 {u_{2}}^{\frac{1}{2}} + C$

Step 14

Simplify.

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Step 14.1

Simplify.

$\frac{1}{25} \cdot \frac{2}{3} {u_{1}}^{\frac{3}{2}} + 2 {u_{2}}^{\frac{1}{2}} + C$

Step 14.2

Simplify.

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Step 14.2.1

Multiply $\frac{1}{25}$ by $\frac{2}{3}$ .

$\frac{2}{25 \cdot 3} {u_{1}}^{\frac{3}{2}} + 2 {u_{2}}^{\frac{1}{2}} + C$

Step 14.2.2

Multiply $25$ by $3$ .

$\frac{2}{75} {u_{1}}^{\frac{3}{2}} + 2 {u_{2}}^{\frac{1}{2}} + C$

Step 15

Substitute back in for each integration substitution variable.

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Step 15.1

Replace all occurrences of $u_{1}$ with $5 x$ .

$\frac{2}{75} {(5 x)}^{\frac{3}{2}} + 2 {u_{2}}^{\frac{1}{2}} + C$

Step 15.2

Replace all occurrences of $u_{2}$ with $5 x$ .

$\frac{2}{75} {(5 x)}^{\frac{3}{2}} + 2 {(5 x)}^{\frac{1}{2}} + C$