Calculus Examples

$\int_{0}^{3} \frac{1}{{(1 - x)}^{2}} d x$

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

Substitute the lower limit in for $x$ in $u = 1 - x$ .

$u_{lower} = 1 - 0$

Step 1.3

Subtract $0$ from $1$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 1 - x$ .

$u_{upper} = 1 - 1 \cdot 3$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 1$ by $3$ .

$u_{upper} = 1 - 3$

Step 1.5.2

Subtract $3$ from $1$ .

$u_{upper} = - 2$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = - 2$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{1}^{- 2} \frac{- 1}{u^{2}} d u$

Step 2

Move the negative in front of the fraction.

$\int_{1}^{- 2} - \frac{1}{u^{2}} d u$

Step 3

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

$- \int_{1}^{- 2} \frac{1}{u^{2}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

$- \int_{1}^{- 2} u^{2 \cdot - 1} d u$

Step 4.2.2

Multiply $2$ by $- 1$ .

$- \int_{1}^{- 2} u^{- 2} d u$

Step 5

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

$- (- u^{- 1}]_{1}^{- 2})$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $- u^{- 1}$ at $- 2$ and at $1$ .

$- ((- {(- 2)}^{- 1}) + 1^{- 1})$

Step 6.2

Simplify.

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Step 6.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 6.2.2

Move the negative in front of the fraction.

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

Step 6.2.3

Multiply $- 1$ by $- 1$ .

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

Step 6.2.4

Multiply $\frac{1}{2}$ by $1$ .

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

Step 6.2.5

One to any power is one.

$- (\frac{1}{2} + 1)$

Step 6.2.6

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

$- (\frac{1}{2} + \frac{2}{2})$

Step 6.2.7

Combine the numerators over the common denominator.

$- \frac{1 + 2}{2}$

Step 6.2.8

Add $1$ and $2$ .

$- \frac{3}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{2}$

Decimal Form:

$- 1.5$

Mixed Number Form:

$- 1 \frac{1}{2}$

Step 8