Calculus Examples

$\int_{0}^{\sqrt{5}} \frac{x}{\sqrt{x^{2} + 4}} d x$

Step 1

Let $u = x^{2} + 4$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x^{2} + 4$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{2} + 4$ .

$\frac{d}{d x} [x^{2} + 4]$

Step 1.1.2

By the Sum Rule, the derivative of $x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [4]$

Step 1.1.4

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$2 x + 0$

Step 1.1.5

Add $2 x$ and $0$ .

$2 x$

Step 1.2

Substitute the lower limit in for $x$ in $u = x^{2} + 4$ .

$u_{lower} = 0^{2} + 4$

Step 1.3

Simplify.

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Step 1.3.1

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 0 + 4$

Step 1.3.2

Add $0$ and $4$ .

$u_{lower} = 4$

Step 1.4

Substitute the upper limit in for $x$ in $u = x^{2} + 4$ .

$u_{upper} = {\sqrt{5}}^{2} + 4$

Step 1.5

Simplify.

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Step 1.5.1

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 1.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$u_{upper} = {(5^{\frac{1}{2}})}^{2} + 4$

Step 1.5.1.2

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

$u_{upper} = 5^{\frac{1}{2} \cdot 2} + 4$

Step 1.5.1.3

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

$u_{upper} = 5^{\frac{2}{2}} + 4$

Step 1.5.1.4

Cancel the common factor of $2$ .

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Step 1.5.1.4.1

Cancel the common factor.

$u_{upper} = 5^{\frac{2}{2}} + 4$

Step 1.5.1.4.2

Rewrite the expression.

$u_{upper} = 5^{1} + 4$

Step 1.5.1.5

Evaluate the exponent.

$u_{upper} = 5 + 4$

Step 1.5.2

Add $5$ and $4$ .

$u_{upper} = 9$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 4$

$u_{upper} = 9$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{4}^{9} \frac{1}{\sqrt{u}} \cdot \frac{1}{2} d u$

Step 2

Simplify.

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Step 2.1

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

$\int_{4}^{9} \frac{1}{\sqrt{u} \cdot 2} d u$

Step 2.2

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

$\int_{4}^{9} \frac{1}{2 \sqrt{u}} d u$

Step 3

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

$\frac{1}{2} \int_{4}^{9} \frac{1}{\sqrt{u}} d u$

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 $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$\frac{1}{2} \int_{4}^{9} \frac{1}{u^{\frac{1}{2}}} d u$

Step 4.2

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

$\frac{1}{2} \int_{4}^{9} {(u^{\frac{1}{2}})}^{- 1} d u$

Step 4.3

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

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Step 4.3.1

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

$\frac{1}{2} \int_{4}^{9} u^{\frac{1}{2} \cdot - 1} d u$

Step 4.3.2

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

$\frac{1}{2} \int_{4}^{9} u^{\frac{- 1}{2}} d u$

Step 4.3.3

Move the negative in front of the fraction.

$\frac{1}{2} \int_{4}^{9} u^{- \frac{1}{2}} d u$

Step 5

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

$\frac{1}{2} 2 u^{\frac{1}{2}}]_{4}^{9}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $2 u^{\frac{1}{2}}$ at $9$ and at $4$ .

$\frac{1}{2} ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 6.2

Simplify.

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Step 6.2.1

Rewrite $9$ as $3^{2}$ .

$\frac{1}{2} (2 \cdot {(3^{2})}^{\frac{1}{2}} - 2 \cdot 4^{\frac{1}{2}})$

Step 6.2.2

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

$\frac{1}{2} (2 \cdot 3^{2 (\frac{1}{2})} - 2 \cdot 4^{\frac{1}{2}})$

Step 6.2.3

Cancel the common factor of $2$ .

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Step 6.2.3.1

Cancel the common factor.

$\frac{1}{2} (2 \cdot 3^{2 (\frac{1}{2})} - 2 \cdot 4^{\frac{1}{2}})$

Step 6.2.3.2

Rewrite the expression.

$\frac{1}{2} (2 \cdot 3^{1} - 2 \cdot 4^{\frac{1}{2}})$

Step 6.2.4

Evaluate the exponent.

$\frac{1}{2} (2 \cdot 3 - 2 \cdot 4^{\frac{1}{2}})$

Step 6.2.5

Multiply $2$ by $3$ .

$\frac{1}{2} (6 - 2 \cdot 4^{\frac{1}{2}})$

Step 6.2.6

Rewrite $4$ as $2^{2}$ .

$\frac{1}{2} (6 - 2 \cdot {(2^{2})}^{\frac{1}{2}})$

Step 6.2.7

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

$\frac{1}{2} (6 - 2 \cdot 2^{2 (\frac{1}{2})})$

Step 6.2.8

Cancel the common factor of $2$ .

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Step 6.2.8.1

Cancel the common factor.

$\frac{1}{2} (6 - 2 \cdot 2^{2 (\frac{1}{2})})$

Step 6.2.8.2

Rewrite the expression.

$\frac{1}{2} (6 - 2 \cdot 2^{1})$

Step 6.2.9

Evaluate the exponent.

$\frac{1}{2} (6 - 2 \cdot 2)$

Step 6.2.10

Multiply $- 2$ by $2$ .

$\frac{1}{2} (6 - 4)$

Step 6.2.11

Subtract $4$ from $6$ .

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

Step 6.2.12

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

$\frac{2}{2}$

Step 6.2.13

Cancel the common factor of $2$ .

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Step 6.2.13.1

Cancel the common factor.

$\frac{2}{2}$

Step 6.2.13.2

Rewrite the expression.

$1$

Step 7