Calculus Examples

$\int \frac{5}{{(1 - x)}^{2}} d x$

Step 1

Let $u = 1 - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 1 - x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Rewrite.

$\frac{- 1}{1}$

Step 1.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 1.2

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

$\int \frac{- 5}{u^{2}} d u$

Step 2

Move the negative in front of the fraction.

$\int - \frac{5}{u^{2}} d u$

Step 3

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

$- \int \frac{5}{u^{2}} d u$

Step 4

Since $5$ is constant with respect to $u$ , move $5$ out of the integral.

$- (5 \int \frac{1}{u^{2}} d u)$

Step 5

Simplify the expression.

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Step 5.1

Multiply $5$ by $- 1$ .

$- 5 \int \frac{1}{u^{2}} d u$

Step 5.2

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

$- 5 \int {(u^{2})}^{- 1} d u$

Step 5.3

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

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Step 5.3.1

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

$- 5 \int u^{2 \cdot - 1} d u$

Step 5.3.2

Multiply $2$ by $- 1$ .

$- 5 \int u^{- 2} d u$

Step 6

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

$- 5 (- u^{- 1} + C)$

Step 7

Simplify.

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Step 7.1

Rewrite $- 5 (- u^{- 1} + C)$ as $- 5 (- \frac{1}{u}) + C$ .

$- 5 (- \frac{1}{u}) + C$

Step 7.2

Simplify.

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Step 7.2.1

Multiply $- 1$ by $- 5$ .

$5 \frac{1}{u} + C$

Step 7.2.2

Combine $5$ and $\frac{1}{u}$ .

$\frac{5}{u} + C$

Step 8

Replace all occurrences of $u$ with $1 - x$ .

$\frac{5}{1 - x} + C$