Calculus Examples

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

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

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

$\int \frac{{\sqrt{u - 4}}^{2}}{u^{\frac{3}{2}}} \cdot \frac{1}{2} d u$

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Cancel the common factor.

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

Step 2.1.4.2

Rewrite the expression.

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

Step 2.1.5

Simplify.

$\int \frac{u - 4}{u^{\frac{3}{2}}} \cdot \frac{1}{2} d u$

Step 2.2

Multiply $\frac{u - 4}{u^{\frac{3}{2}}}$ by $\frac{1}{2}$ .

$\int \frac{u - 4}{u^{\frac{3}{2}} \cdot 2} d u$

Step 2.3

Move $2$ to the left of $u^{\frac{3}{2}}$ .

$\int \frac{u - 4}{2 u^{\frac{3}{2}}} d u$

Step 3

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

$\frac{1}{2} \int \frac{u - 4}{u^{\frac{3}{2}}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 4.2.2.1

Combine $\frac{3}{2}$ and $- 1$ .

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

Step 4.2.2.2

Multiply $3$ by $- 1$ .

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

Step 4.2.3

Move the negative in front of the fraction.

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

Step 5

Expand $(u - 4) u^{- \frac{3}{2}}$ .

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Step 5.1

Apply the distributive property.

$\frac{1}{2} \int u \cdot u^{- \frac{3}{2}} - 4 u^{- \frac{3}{2}} d u$

Step 5.2

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

$\frac{1}{2} \int u^{1} u^{- \frac{3}{2}} - 4 u^{- \frac{3}{2}} d u$

Step 5.3

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

$\frac{1}{2} \int u^{1 - \frac{3}{2}} - 4 u^{- \frac{3}{2}} d u$

Step 5.4

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

$\frac{1}{2} \int u^{\frac{2}{2} - \frac{3}{2}} - 4 u^{- \frac{3}{2}} d u$

Step 5.5

Combine the numerators over the common denominator.

$\frac{1}{2} \int u^{\frac{2 - 3}{2}} - 4 u^{- \frac{3}{2}} d u$

Step 5.6

Subtract $3$ from $2$ .

$\frac{1}{2} \int u^{\frac{- 1}{2}} - 4 u^{- \frac{3}{2}} d u$

Step 6

Move the negative in front of the fraction.

$\frac{1}{2} \int u^{- \frac{1}{2}} - 4 u^{- \frac{3}{2}} d u$

Step 7

Split the single integral into multiple integrals.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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

Step 12

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

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

Step 13

Simplify.

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Step 13.1

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

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

Step 13.2

Combine the numerators over the common denominator.

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

Step 13.3

Simplify the numerator.

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Step 13.3.1

Factor $2$ out of $2 {(x^{2} + 4)}^{\frac{1}{2}} {(x^{2} + 4)}^{\frac{1}{2}} + 8$ .

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Step 13.3.1.1

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

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

Step 13.3.1.2

Factor $2$ out of $8$ .

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

Step 13.3.1.3

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

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

Step 13.3.2

Multiply ${(x^{2} + 4)}^{\frac{1}{2}}$ by ${(x^{2} + 4)}^{\frac{1}{2}}$ by adding the exponents.

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Step 13.3.2.1

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

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

Step 13.3.2.2

Combine the numerators over the common denominator.

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

Step 13.3.2.3

Add $1$ and $1$ .

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

Step 13.3.2.4

Divide $2$ by $2$ .

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

Step 13.3.3

Simplify ${(x^{2} + 4)}^{1}$ .

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

Step 13.3.4

Add $4$ and $4$ .

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

Step 13.4

Combine.

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

Step 13.5

Cancel the common factor.

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

Step 13.6

Rewrite the expression.

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

Step 13.7

Multiply $x^{2} + 8$ by $1$ .

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