Calculus Examples

$\int \frac{2}{{(6 - x)}^{\frac{3}{2}}} d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int \frac{1}{{(6 - x)}^{\frac{3}{2}}} d x$

Step 2

Let $u = 6 - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 6 - x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Rewrite.

$\frac{- 1}{1}$

Step 2.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 2.2

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

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

Step 3

Move the negative in front of the fraction.

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

Step 4

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 5

Simplify the expression.

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Step 5.1

Multiply $- 1$ by $2$ .

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

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

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Step 5.3.2.1

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

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

Step 5.3.2.2

Multiply $3$ by $- 1$ .

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

Step 5.3.3

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

Move the negative in front of the fraction.

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

Step 7.2.3

Multiply $- 1$ by $- 2$ .

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

Step 7.2.4

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

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

Step 7.2.5

Multiply $2$ by $2$ .

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

Step 8

Replace all occurrences of $u$ with $6 - x$ .

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