Calculus Examples

$\int_{- 1}^{1} \frac{5 r}{{(4 + r^{2})}^{2}} d r$

Step 1

Let $u = 4 + r^{2}$ . Then $d u = 2 r d r$ , so $\frac{1}{2} d u = r d r$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 4 + r^{2}$ . Find $\frac{d u}{d r}$ .

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Step 1.1.1

Differentiate $4 + r^{2}$ .

$\frac{d}{d r} [4 + r^{2}]$

Step 1.1.2

By the Sum Rule, the derivative of $4 + r^{2}$ with respect to $r$ is $\frac{d}{d r} [4] + \frac{d}{d r} [r^{2}]$ .

$\frac{d}{d r} [4] + \frac{d}{d r} [r^{2}]$

Step 1.1.3

Since $4$ is constant with respect to $r$ , the derivative of $4$ with respect to $r$ is $0$ .

$0 + \frac{d}{d r} [r^{2}]$

Step 1.1.4

Differentiate using the Power Rule which states that $\frac{d}{d r} [r^{n}]$ is $n r^{n - 1}$ where $n = 2$ .

$0 + 2 r$

Step 1.1.5

Add $0$ and $2 r$ .

$2 r$

Step 1.2

Substitute the lower limit in for $r$ in $u = 4 + r^{2}$ .

$u_{lower} = 4 + {(- 1)}^{2}$

Step 1.3

Simplify.

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Step 1.3.1

Raise $- 1$ to the power of $2$ .

$u_{lower} = 4 + 1$

Step 1.3.2

Add $4$ and $1$ .

$u_{lower} = 5$

Step 1.4

Substitute the upper limit in for $r$ in $u = 4 + r^{2}$ .

$u_{upper} = 4 + 1^{2}$

Step 1.5

Simplify.

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Step 1.5.1

One to any power is one.

$u_{upper} = 4 + 1$

Step 1.5.2

Add $4$ and $1$ .

$u_{upper} = 5$

Step 1.6

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

$u_{lower} = 5$

$u_{upper} = 5$

Step 1.7

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

$\int_{5}^{5} \frac{5}{u^{2}} \cdot \frac{1}{2} d u$

Step 2

Simplify.

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Step 2.1

Multiply $\frac{5}{u^{2}}$ by $\frac{1}{2}$ .

$\int_{5}^{5} \frac{5}{u^{2} \cdot 2} d u$

Step 2.2

Move $2$ to the left of $u^{2}$ .

$\int_{5}^{5} \frac{5}{2 u^{2}} d u$

Step 3

Since $\frac{5}{2}$ is constant with respect to $u$ , move $\frac{5}{2}$ out of the integral.

$\frac{5}{2} \int_{5}^{5} \frac{1}{u^{2}} d u$

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.

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.2.2

Multiply $2$ by $- 1$ .

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

Step 5

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

$\frac{5}{2} - u^{- 1}]_{5}^{5}$

Step 6

Evaluate $- u^{- 1}$ at $5$ and at $5$ .

$\frac{5}{2} ((- 5^{- 1}) + 5^{- 1})$

Step 7

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

$\frac{5}{2} (- \frac{1}{5} + 5^{- 1})$

Step 8

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

$\frac{5}{2} (- \frac{1}{5} + \frac{1}{5})$

Step 9

Add $- \frac{1}{5}$ and $\frac{1}{5}$ .

$\frac{5}{2} \cdot 0$

Step 10

Multiply $\frac{5}{2}$ by $0$ .

$0$