Calculus Examples

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

Step 1

Simplify.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Since $d$ is constant with respect to $x$ , move $d$ out of the integral.

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

Step 3

Let $u = x^{2} + 2$ . 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 3.1

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

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Step 3.1.1

Differentiate $x^{2} + 2$ .

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$2 x + 0$

Step 3.1.5

Add $2 x$ and $0$ .

$2 x$

Step 3.2

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

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

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

Step 6

Combine fractions.

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Step 6.1

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

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

Step 6.2

Apply basic rules of exponents.

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Step 6.2.1

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

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

Step 6.2.2

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

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Step 6.2.2.1

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

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

Step 6.2.2.2

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

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Step 6.2.2.2.1

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

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

Step 6.2.2.2.2

Multiply $3$ by $- 1$ .

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

Step 6.2.2.3

Move the negative in front of the fraction.

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Rewrite $\frac{d}{2} (- 2 u^{- \frac{1}{2}} + C)$ as $\frac{d}{2} (- 2 \frac{1}{u^{\frac{1}{2}}}) + C$ .

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

Step 8.2

Simplify.

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Step 8.2.1

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

$\frac{d}{2} \cdot \frac{- 2}{u^{\frac{1}{2}}} + C$

Step 8.2.2

Move the negative in front of the fraction.

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

Step 8.2.3

Multiply $\frac{d}{2}$ by $\frac{2}{u^{\frac{1}{2}}}$ .

$- \frac{d \cdot 2}{2 u^{\frac{1}{2}}} + C$

Step 8.2.4

Move $2$ to the left of $d$ .

$- \frac{2 \cdot d}{2 u^{\frac{1}{2}}} + C$

Step 8.2.5

Cancel the common factor.

$- \frac{2 d}{2 u^{\frac{1}{2}}} + C$

Step 8.2.6

Rewrite the expression.

$- \frac{d}{u^{\frac{1}{2}}} + C$

Step 9

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

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