Calculus Examples

$\int_{0}^{4} \frac{1}{{(x - 2)}^{2}} d x$

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

Substitute the lower limit in for $x$ in $u = x - 2$ .

$u_{lower} = 0 - 2$

Step 1.3

Subtract $2$ from $0$ .

$u_{lower} = - 2$

Step 1.4

Substitute the upper limit in for $x$ in $u = x - 2$ .

$u_{upper} = 4 - 2$

Step 1.5

Subtract $2$ from $4$ .

$u_{upper} = 2$

Step 1.6

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

$u_{lower} = - 2$

$u_{upper} = 2$

Step 1.7

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

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

Step 2

Apply basic rules of exponents.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $- 1$ .

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

Step 3

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

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

Step 4

Substitute and simplify.

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Step 4.1

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

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

Step 4.2

Simplify.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

Move the negative in front of the fraction.

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

Step 4.2.4

Subtract $\frac{1}{2}$ from $- \frac{1}{2}$ .

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

Step 4.2.5

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

$\frac{- 2}{2}$

Step 4.2.6

Cancel the common factor of $- 2$ and $2$ .

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Step 4.2.6.1

Factor $2$ out of $- 2$ .

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

Step 4.2.6.2

Cancel the common factors.

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Step 4.2.6.2.1

Factor $2$ out of $2$ .

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

Step 4.2.6.2.2

Cancel the common factor.

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

Step 4.2.6.2.3

Rewrite the expression.

$\frac{- 1}{1}$

Step 4.2.6.2.4

Divide $- 1$ by $1$ .

$- 1$

Step 5