Calculus Examples

$\int \frac{x^{3}}{\sqrt{4 + x^{2}}} d x$

Step 1

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

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Step 1.1.1

Differentiate $4 + x^{2}$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

$0 + \frac{d}{d x} [x^{2}]$

Step 1.1.4

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

$0 + 2 x$

Step 1.1.5

Add $0$ and $2 x$ .

$2 x$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

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

Step 2

Simplify.

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Step 2.1

Rewrite ${\sqrt{u - 4}}^{2}$ as $u - 4$ .

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u - 4}$ as ${(u - 4)}^{\frac{1}{2}}$ .

$\int \frac{{({(u - 4)}^{\frac{1}{2}})}^{2}}{\sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.1.2

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

$\int \frac{{(u - 4)}^{\frac{1}{2} \cdot 2}}{\sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.1.3

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

$\int \frac{{(u - 4)}^{\frac{2}{2}}}{\sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Cancel the common factor.

$\int \frac{{(u - 4)}^{\frac{2}{2}}}{\sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.1.4.2

Rewrite the expression.

$\int \frac{{(u - 4)}^{1}}{\sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.1.5

Simplify.

$\int \frac{u - 4}{\sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.2

Multiply $\frac{u - 4}{\sqrt{u}}$ by $\frac{1}{2}$ .

$\int \frac{u - 4}{\sqrt{u} \cdot 2} d u$

Step 2.3

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

$\int \frac{u - 4}{2 \sqrt{u}} d u$

Step 3

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

$\frac{1}{2} \int \frac{u - 4}{\sqrt{u}} d u$

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

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

Step 4.2

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{2} \int (u - 4) u^{\frac{1}{2} \cdot - 1} d u$

Step 4.3.2

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

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

Step 4.3.3

Move the negative in front of the fraction.

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

Step 5

Expand $(u - 4) u^{- \frac{1}{2}}$ .

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Step 5.1

Apply the distributive property.

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

Step 5.2

Raise $u$ to the power of $1$ .

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Combine the numerators over the common denominator.

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

Step 5.6

Subtract $1$ from $2$ .

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

$\frac{1}{2} (\frac{2}{3} u^{\frac{3}{2}} + C + \int - 4 u^{- \frac{1}{2}} d u)$

Step 8

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

$\frac{1}{2} (\frac{2}{3} u^{\frac{3}{2}} + C - 4 \int u^{- \frac{1}{2}} d u)$

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

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

Step 10.2

Reorder terms.

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

Step 11

Replace all occurrences of $u$ with $4 + x^{2}$ .

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

Step 12

Simplify.

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Step 12.1

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

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

Step 12.2

To write $- 8 {(4 + x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 12.3

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

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

Step 12.4

Combine the numerators over the common denominator.

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

Step 12.5

Multiply $3$ by $- 8$ .

$\frac{1}{2} \cdot \frac{2 {(4 + x^{2})}^{\frac{3}{2}} - 24 {(4 + x^{2})}^{\frac{1}{2}}}{3} + C$

Step 12.6

Multiply $\frac{1}{2} \cdot \frac{2 {(4 + x^{2})}^{\frac{3}{2}} - 24 {(4 + x^{2})}^{\frac{1}{2}}}{3}$ .

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Step 12.6.1

Multiply $\frac{1}{2}$ by $\frac{2 {(4 + x^{2})}^{\frac{3}{2}} - 24 {(4 + x^{2})}^{\frac{1}{2}}}{3}$ .

$\frac{2 {(4 + x^{2})}^{\frac{3}{2}} - 24 {(4 + x^{2})}^{\frac{1}{2}}}{2 \cdot 3} + C$

Step 12.6.2

Multiply $2$ by $3$ .

$\frac{2 {(4 + x^{2})}^{\frac{3}{2}} - 24 {(4 + x^{2})}^{\frac{1}{2}}}{6} + C$

Step 13

Reorder terms.

$\frac{1}{6} (2 {(4 + x^{2})}^{\frac{3}{2}} - 24 {(4 + x^{2})}^{\frac{1}{2}}) + C$