Calculus Examples

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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Step 2.1

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

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

Step 2.2

Move $\sqrt{x}$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Add $1$ and $1$ .

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

Step 2.7

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

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Step 2.7.1

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

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

Step 2.7.2

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

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

Step 2.7.3

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

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

Step 2.7.4

Cancel the common factor of $2$ .

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Step 2.7.4.1

Cancel the common factor.

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

Step 2.7.4.2

Rewrite the expression.

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

Step 2.7.5

Simplify.

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

Step 3

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 3.1

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

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Step 3.1.1

Differentiate $2 x$ .

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

Step 3.1.2

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 3.1.3

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

$2 \cdot 1$

Step 3.1.4

Multiply $2$ by $1$ .

$2$

Step 3.2

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Move $2$ to the left of $u_{1}$ .

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

Step 5

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

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

Step 6

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 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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 6.1.3

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 6.1.4

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

$\frac{1}{2}$

Step 6.2

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

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

Step 7

Simplify the expression.

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Step 7.1

Simplify.

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Step 7.1.1

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

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

Step 7.1.2

Multiply $2$ by $1$ .

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

Step 7.1.3

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

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

Step 7.1.4

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

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

Step 7.1.5

Cancel the common factor of $2$ .

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Step 7.1.5.1

Cancel the common factor.

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

Step 7.1.5.2

Rewrite the expression.

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

Step 7.2

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

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

Step 7.3

Simplify.

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Step 7.3.1

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

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

Step 7.3.2

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

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Step 7.3.2.1

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

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Step 7.3.2.1.1

Raise $u_{2}$ to the power of $1$ .

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

Step 7.3.2.1.2

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

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

Step 7.3.2.2

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

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

Step 7.3.2.3

Combine the numerators over the common denominator.

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

Step 7.3.2.4

Subtract $1$ from $2$ .

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

Step 7.4

Apply basic rules of exponents.

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Step 7.4.1

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

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

Step 7.4.2

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

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Step 7.4.2.1

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

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

Step 7.4.2.2

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

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

Step 7.4.2.3

Move the negative in front of the fraction.

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Rewrite $\frac{1}{2} (2 {u_{2}}^{\frac{1}{2}} + C)$ as $\frac{1}{2} \cdot 2 {u_{2}}^{\frac{1}{2}} + C$ .

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

Step 9.2

Rewrite $\frac{1}{2} \cdot 2 {u_{2}}^{\frac{1}{2}} + C$ as $1 {u_{2}}^{\frac{1}{2}} + C$ .

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

Step 9.3

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

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

Step 10

Substitute back in for each integration substitution variable.

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Step 10.1

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

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

Step 10.2

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

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

Step 11

Simplify.

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Step 11.1

Reduce the expression $\frac{2 x}{2}$ by cancelling the common factors.

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Step 11.1.1

Cancel the common factor.

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

Step 11.1.2

Rewrite the expression.

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

Step 11.2

Divide $x$ by $1$ .

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