Calculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

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

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

Step 1.2.2

Move $\sqrt{x}$ .

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

Step 1.2.3

Raise $\sqrt{x}$ to the power of $1$ .

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

Step 1.2.4

Raise $\sqrt{x}$ to the power of $1$ .

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

Step 1.2.5

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

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

Step 1.2.6

Add $1$ and $1$ .

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

Step 1.2.7

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 1.2.7.1

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

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

Step 1.2.7.2

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

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

Step 1.2.7.3

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

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

Step 1.2.7.4

Cancel the common factor of $2$ .

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Step 1.2.7.4.1

Cancel the common factor.

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

Step 1.2.7.4.2

Rewrite the expression.

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

Step 1.2.7.5

Simplify.

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

Step 2

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

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Step 2.1

Let $u_{1} = 2 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 2.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 2.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 2.1.3

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

$2 \cdot 1$

Step 2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2

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

$\int (\sqrt{\frac{u_{1}}{2}} + \frac{\sqrt{\frac{u_{1}}{2}}}{u_{1}}) \frac{1}{2} d u_{1}$

Step 3

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

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

Step 4

Split the single integral into multiple integrals.

$\frac{1}{2} (\int \sqrt{\frac{u_{1}}{2}} d u_{1} + \int \frac{\sqrt{\frac{u_{1}}{2}}}{u_{1}} d u_{1})$

Step 5

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

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Step 5.1

Let $u_{2} = \frac{u_{1}}{2}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 5.1.1

Differentiate $\frac{u_{1}}{2}$ .

$\frac{d}{d u_{1}} [\frac{u_{1}}{2}]$

Step 5.1.2

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , the derivative of $\frac{u_{1}}{2}$ with respect to $u_{1}$ is $\frac{1}{2} \frac{d}{d u_{1}} [u_{1}]$ .

$\frac{1}{2} \frac{d}{d u_{1}} [u_{1}]$

Step 5.1.3

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

$\frac{1}{2} \cdot 1$

Step 5.1.4

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

$\frac{1}{2}$

Step 5.2

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

$\frac{1}{2} (\int \sqrt{u_{2}} \frac{1}{\frac{1}{2}} d u_{2} + \int \frac{\sqrt{\frac{u_{1}}{2}}}{u_{1}} d u_{1})$

Step 6

Simplify.

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Step 6.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\frac{1}{2} (\int \sqrt{u_{2}} (1 \cdot 2) d u_{2} + \int \frac{\sqrt{\frac{u_{1}}{2}}}{u_{1}} d u_{1})$

Step 6.2

Multiply $2$ by $1$ .

$\frac{1}{2} (\int \sqrt{u_{2}} \cdot 2 d u_{2} + \int \frac{\sqrt{\frac{u_{1}}{2}}}{u_{1}} d u_{1})$

Step 6.3

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

$\frac{1}{2} (\int 2 \sqrt{u_{2}} d u_{2} + \int \frac{\sqrt{\frac{u_{1}}{2}}}{u_{1}} d u_{1})$

Step 7

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

$\frac{1}{2} (2 \int \sqrt{u_{2}} d u_{2} + \int \frac{\sqrt{\frac{u_{1}}{2}}}{u_{1}} d u_{1})$

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Let $u_{3} = \frac{u_{1}}{2}$ . Then $d u_{3} = \frac{1}{2} d u_{1}$ , so $2 d u_{3} = d u_{1}$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 11.1

Let $u_{3} = \frac{u_{1}}{2}$ . Find $\frac{d u_{3}}{d u_{1}}$ .

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Step 11.1.1

Differentiate $\frac{u_{1}}{2}$ .

$\frac{d}{d u_{1}} [\frac{u_{1}}{2}]$

Step 11.1.2

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , the derivative of $\frac{u_{1}}{2}$ with respect to $u_{1}$ is $\frac{1}{2} \frac{d}{d u_{1}} [u_{1}]$ .

$\frac{1}{2} \frac{d}{d u_{1}} [u_{1}]$

Step 11.1.3

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

$\frac{1}{2} \cdot 1$

Step 11.1.4

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

$\frac{1}{2}$

Step 11.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

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

Step 12

Simplify the expression.

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Step 12.1

Simplify.

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Step 12.1.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

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

Step 12.1.2

Multiply $2$ by $1$ .

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

Step 12.1.3

Combine $\frac{\sqrt{u_{3}}}{2 u_{3}}$ and $2$ .

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

Step 12.1.4

Move $2$ to the left of $\sqrt{u_{3}}$ .

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

Step 12.1.5

Cancel the common factor of $2$ .

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Step 12.1.5.1

Cancel the common factor.

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

Step 12.1.5.2

Rewrite the expression.

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

Step 12.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u_{3}}$ as ${u_{3}}^{\frac{1}{2}}$ .

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

Step 12.3

Simplify.

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Step 12.3.1

Move ${u_{3}}^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 12.3.2

Multiply $u_{3}$ by ${u_{3}}^{- \frac{1}{2}}$ by adding the exponents.

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Step 12.3.2.1

Multiply $u_{3}$ by ${u_{3}}^{- \frac{1}{2}}$ .

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Step 12.3.2.1.1

Raise $u_{3}$ to the power of $1$ .

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

Step 12.3.2.1.2

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

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

Step 12.3.2.2

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

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

Step 12.3.2.3

Combine the numerators over the common denominator.

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

Step 12.3.2.4

Subtract $1$ from $2$ .

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

Step 12.4

Apply basic rules of exponents.

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Step 12.4.1

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

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

Step 12.4.2

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

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Step 12.4.2.1

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

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

Step 12.4.2.2

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

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

Step 12.4.2.3

Move the negative in front of the fraction.

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

Step 13

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

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

Step 14

Simplify.

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Step 14.1

Simplify.

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

Step 14.2

Reorder terms.

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

Step 15

Substitute back in for each integration substitution variable.

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Step 15.1

Replace all occurrences of $u_{2}$ with $\frac{u_{1}}{2}$ .

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

Step 15.2

Replace all occurrences of $u_{3}$ with $\frac{u_{1}}{2}$ .

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

Step 15.3

Replace all occurrences of $u_{1}$ with $2 x$ .

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

Step 16

Simplify.

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Step 16.1

Simplify each term.

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Step 16.1.1

Cancel the common factor of $2$ .

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Step 16.1.1.1

Cancel the common factor.

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

Step 16.1.1.2

Divide $x$ by $1$ .

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

Step 16.1.2

Combine $\frac{4}{3}$ and $x^{\frac{3}{2}}$ .

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

Step 16.1.3

Cancel the common factor of $2$ .

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Step 16.1.3.1

Cancel the common factor.

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

Step 16.1.3.2

Divide $x$ by $1$ .

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

Step 16.2

Apply the distributive property.

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

Step 16.3

Combine.

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

Step 16.4

Cancel the common factor of $2$ .

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Step 16.4.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

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

Step 16.4.2

Cancel the common factor.

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

Step 16.4.3

Rewrite the expression.

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

Step 16.5

Simplify each term.

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Step 16.5.1

Factor $2$ out of $1 (4 x^{\frac{3}{2}})$ .

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

Step 16.5.2

Cancel the common factors.

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Step 16.5.2.1

Factor $2$ out of $2 \cdot 3$ .

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

Step 16.5.2.2

Cancel the common factor.

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

Step 16.5.2.3

Rewrite the expression.

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

Step 16.5.3

Multiply $2$ by $1$ .

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

Step 17

Reorder terms.

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