Calculus Examples

$\int_{- 8}^{4} \frac{14}{{(9 - 2 x)}^{\frac{3}{2}}} d x$

Step 1

Since $14$ is constant with respect to $x$ , move $14$ out of the integral.

$14 \int_{- 8}^{4} \frac{1}{{(9 - 2 x)}^{\frac{3}{2}}} d x$

Step 2

Let $u = 9 - 2 x$ . Then $d u = - 2 d x$ , so $- \frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 9 - 2 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $9 - 2 x$ .

$\frac{d}{d x} [9 - 2 x]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $9 - 2 x$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- 2 x]$ .

$\frac{d}{d x} [9] + \frac{d}{d x} [- 2 x]$

Step 2.1.2.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 2 x]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 2.1.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$0 - 2 \frac{d}{d x} [x]$

Step 2.1.3.2

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

$0 - 2 \cdot 1$

Step 2.1.3.3

Multiply $- 2$ by $1$ .

$0 - 2$

Step 2.1.4

Subtract $2$ from $0$ .

$- 2$

Step 2.2

Substitute the lower limit in for $x$ in $u = 9 - 2 x$ .

$u_{lower} = 9 - 2 \cdot - 8$

Step 2.3

Simplify.

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Step 2.3.1

Multiply $- 2$ by $- 8$ .

$u_{lower} = 9 + 16$

Step 2.3.2

Add $9$ and $16$ .

$u_{lower} = 25$

Step 2.4

Substitute the upper limit in for $x$ in $u = 9 - 2 x$ .

$u_{upper} = 9 - 2 \cdot 4$

Step 2.5

Simplify.

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Step 2.5.1

Multiply $- 2$ by $4$ .

$u_{upper} = 9 - 8$

Step 2.5.2

Subtract $8$ from $9$ .

$u_{upper} = 1$

Step 2.6

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

$u_{lower} = 25$

$u_{upper} = 1$

Step 2.7

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

$14 \int_{25}^{1} \frac{1}{u^{\frac{3}{2}}} \cdot \frac{1}{- 2} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$14 \int_{25}^{1} \frac{1}{u^{\frac{3}{2}}} (- \frac{1}{2}) d u$

Step 3.2

Multiply $\frac{1}{u^{\frac{3}{2}}}$ by $\frac{1}{2}$ .

$14 \int_{25}^{1} - \frac{1}{u^{\frac{3}{2}} \cdot 2} d u$

Step 3.3

Move $2$ to the left of $u^{\frac{3}{2}}$ .

$14 \int_{25}^{1} - \frac{1}{2 u^{\frac{3}{2}}} d u$

Step 4

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

$14 (- \int_{25}^{1} \frac{1}{2 u^{\frac{3}{2}}} d u)$

Step 5

Multiply $- 1$ by $14$ .

$- 14 \int_{25}^{1} \frac{1}{2 u^{\frac{3}{2}}} d u$

Step 6

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$- 14 (\frac{1}{2} \int_{25}^{1} \frac{1}{u^{\frac{3}{2}}} d u)$

Step 7

Simplify the expression.

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Step 7.1

Simplify.

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Step 7.1.1

Combine $\frac{1}{2}$ and $- 14$ .

$\frac{- 14}{2} \int_{25}^{1} \frac{1}{u^{\frac{3}{2}}} d u$

Step 7.1.2

Cancel the common factor of $- 14$ and $2$ .

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Step 7.1.2.1

Factor $2$ out of $- 14$ .

$\frac{2 \cdot - 7}{2} \int_{25}^{1} \frac{1}{u^{\frac{3}{2}}} d u$

Step 7.1.2.2

Cancel the common factors.

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Step 7.1.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot - 7}{2 (1)} \int_{25}^{1} \frac{1}{u^{\frac{3}{2}}} d u$

Step 7.1.2.2.2

Cancel the common factor.

$\frac{2 \cdot - 7}{2 \cdot 1} \int_{25}^{1} \frac{1}{u^{\frac{3}{2}}} d u$

Step 7.1.2.2.3

Rewrite the expression.

$\frac{- 7}{1} \int_{25}^{1} \frac{1}{u^{\frac{3}{2}}} d u$

Step 7.1.2.2.4

Divide $- 7$ by $1$ .

$- 7 \int_{25}^{1} \frac{1}{u^{\frac{3}{2}}} d u$

Step 7.2

Apply basic rules of exponents.

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Step 7.2.1

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

$- 7 \int_{25}^{1} {(u^{\frac{3}{2}})}^{- 1} d u$

Step 7.2.2

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

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Step 7.2.2.1

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

$- 7 \int_{25}^{1} u^{\frac{3}{2} \cdot - 1} d u$

Step 7.2.2.2

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

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Step 7.2.2.2.1

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

$- 7 \int_{25}^{1} u^{\frac{3 \cdot - 1}{2}} d u$

Step 7.2.2.2.2

Multiply $3$ by $- 1$ .

$- 7 \int_{25}^{1} u^{\frac{- 3}{2}} d u$

Step 7.2.2.3

Move the negative in front of the fraction.

$- 7 \int_{25}^{1} u^{- \frac{3}{2}} d u$

Step 8

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

$- 7 (- 2 u^{- \frac{1}{2}}]_{25}^{1})$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $- 2 u^{- \frac{1}{2}}$ at $1$ and at $25$ .

$- 7 ((- 2 \cdot 1^{- \frac{1}{2}}) + 2 \cdot 25^{- \frac{1}{2}})$

Step 9.2

Simplify.

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Step 9.2.1

One to any power is one.

$- 7 (- 2 \cdot 1 + 2 \cdot 25^{- \frac{1}{2}})$

Step 9.2.2

Multiply $- 2$ by $1$ .

$- 7 (- 2 + 2 \cdot 25^{- \frac{1}{2}})$

Step 9.2.3

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

$- 7 (- 2 + 2 \cdot \frac{1}{25^{\frac{1}{2}}})$

Step 9.2.4

Rewrite $25$ as $5^{2}$ .

$- 7 (- 2 + 2 \cdot \frac{1}{{(5^{2})}^{\frac{1}{2}}})$

Step 9.2.5

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

$- 7 (- 2 + 2 \cdot \frac{1}{5^{2 (\frac{1}{2})}})$

Step 9.2.6

Cancel the common factor of $2$ .

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Step 9.2.6.1

Cancel the common factor.

$- 7 (- 2 + 2 \cdot \frac{1}{5^{2 (\frac{1}{2})}})$

Step 9.2.6.2

Rewrite the expression.

$- 7 (- 2 + 2 \cdot \frac{1}{5^{1}})$

Step 9.2.7

Evaluate the exponent.

$- 7 (- 2 + 2 \cdot \frac{1}{5})$

Step 9.2.8

Combine $2$ and $\frac{1}{5}$ .

$- 7 (- 2 + \frac{2}{5})$

Step 9.2.9

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- 7 (- 2 \cdot \frac{5}{5} + \frac{2}{5})$

Step 9.2.10

Combine $- 2$ and $\frac{5}{5}$ .

$- 7 (\frac{- 2 \cdot 5}{5} + \frac{2}{5})$

Step 9.2.11

Combine the numerators over the common denominator.

$- 7 \frac{- 2 \cdot 5 + 2}{5}$

Step 9.2.12

Simplify the numerator.

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Step 9.2.12.1

Multiply $- 2$ by $5$ .

$- 7 \frac{- 10 + 2}{5}$

Step 9.2.12.2

Add $- 10$ and $2$ .

$- 7 (\frac{- 8}{5})$

Step 9.2.13

Move the negative in front of the fraction.

$- 7 (- \frac{8}{5})$

Step 9.2.14

Multiply $- 1$ by $- 7$ .

$7 (\frac{8}{5})$

Step 9.2.15

Combine $7$ and $\frac{8}{5}$ .

$\frac{7 \cdot 8}{5}$

Step 9.2.16

Multiply $7$ by $8$ .

$\frac{56}{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{56}{5}$

Decimal Form:

$11.2$

Mixed Number Form:

$11 \frac{1}{5}$

Step 11