Calculus Examples

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

Step 1

Let $u = x - 2$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $x - 2$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

$1 + \frac{d}{d x} [- 2]$

Step 1.1.4

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

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

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

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

Step 2

Apply basic rules of exponents.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

Multiply $3$ by $- 1$ .

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

Step 2.2.3

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Simplify.

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Step 4.2.1

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

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

Step 4.2.2

Move the negative in front of the fraction.

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

Step 5

Replace all occurrences of $u$ with $x - 2$ .

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