Calculus Examples

$\int_{1}^{2} \frac{1}{{(3 - 5 x)}^{2}} d x$

Step 1

Let $u = 3 - 5 x$ . Then $d u = - 5 d x$ , so $- \frac{1}{5} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 3 - 5 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $3 - 5 x$ .

$\frac{d}{d x} [3 - 5 x]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $3 - 5 x$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- 5 x]$ .

$\frac{d}{d x} [3] + \frac{d}{d x} [- 5 x]$

Step 1.1.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 5 x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 5 x]$ .

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Step 1.1.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$0 - 5 \frac{d}{d x} [x]$

Step 1.1.3.2

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

$0 - 5 \cdot 1$

Step 1.1.3.3

Multiply $- 5$ by $1$ .

$0 - 5$

Step 1.1.4

Subtract $5$ from $0$ .

$- 5$

Step 1.2

Substitute the lower limit in for $x$ in $u = 3 - 5 x$ .

$u_{lower} = 3 - 5 \cdot 1$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $- 5$ by $1$ .

$u_{lower} = 3 - 5$

Step 1.3.2

Subtract $5$ from $3$ .

$u_{lower} = - 2$

Step 1.4

Substitute the upper limit in for $x$ in $u = 3 - 5 x$ .

$u_{upper} = 3 - 5 \cdot 2$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 5$ by $2$ .

$u_{upper} = 3 - 10$

Step 1.5.2

Subtract $10$ from $3$ .

$u_{upper} = - 7$

Step 1.6

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

$u_{lower} = - 2$

$u_{upper} = - 7$

Step 1.7

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

$\int_{- 2}^{- 7} \frac{1}{u^{2}} \cdot \frac{1}{- 5} d u$

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

$\int_{- 2}^{- 7} \frac{1}{u^{2}} (- \frac{1}{5}) d u$

Step 2.2

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

$\int_{- 2}^{- 7} - \frac{1}{u^{2} \cdot 5} d u$

Step 2.3

Move $5$ to the left of $u^{2}$ .

$\int_{- 2}^{- 7} - \frac{1}{5 u^{2}} d u$

Step 3

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

$- \int_{- 2}^{- 7} \frac{1}{5 u^{2}} d u$

Step 4

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$- (\frac{1}{5} \int_{- 2}^{- 7} \frac{1}{u^{2}} d u)$

Step 5

Apply basic rules of exponents.

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Step 5.1

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

$- \frac{1}{5} \int_{- 2}^{- 7} {(u^{2})}^{- 1} d u$

Step 5.2

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

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Step 5.2.1

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

$- \frac{1}{5} \int_{- 2}^{- 7} u^{2 \cdot - 1} d u$

Step 5.2.2

Multiply $2$ by $- 1$ .

$- \frac{1}{5} \int_{- 2}^{- 7} u^{- 2} d u$

Step 6

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

$- \frac{1}{5} - u^{- 1}]_{- 2}^{- 7}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $- u^{- 1}$ at $- 7$ and at $- 2$ .

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

Move the negative in front of the fraction.

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

Step 7.2.3

Multiply $- 1$ by $- 1$ .

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

Step 7.2.4

Multiply $\frac{1}{7}$ by $1$ .

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

Step 7.2.5

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

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

Step 7.2.6

Move the negative in front of the fraction.

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

Step 7.2.7

To write $\frac{1}{7}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 7.2.8

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

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

Step 7.2.9

Write each expression with a common denominator of $14$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.9.1

Multiply $\frac{1}{7}$ by $\frac{2}{2}$ .

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

Step 7.2.9.2

Multiply $7$ by $2$ .

$- \frac{1}{5} (\frac{2}{14} - \frac{1}{2} \cdot \frac{7}{7})$

Step 7.2.9.3

Multiply $\frac{1}{2}$ by $\frac{7}{7}$ .

$- \frac{1}{5} (\frac{2}{14} - \frac{7}{2 \cdot 7})$

Step 7.2.9.4

Multiply $2$ by $7$ .

$- \frac{1}{5} (\frac{2}{14} - \frac{7}{14})$

Step 7.2.10

Combine the numerators over the common denominator.

$- \frac{1}{5} \cdot \frac{2 - 7}{14}$

Step 7.2.11

Subtract $7$ from $2$ .

$- \frac{1}{5} \cdot \frac{- 5}{14}$

Step 7.2.12

Move the negative in front of the fraction.

$- \frac{1}{5} (- \frac{5}{14})$

Step 7.2.13

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{5}) \frac{5}{14}$

Step 7.2.14

Multiply $\frac{1}{5}$ by $1$ .

$\frac{1}{5} \cdot \frac{5}{14}$

Step 7.2.15

Multiply $\frac{1}{5}$ by $\frac{5}{14}$ .

$\frac{5}{5 \cdot 14}$

Step 7.2.16

Multiply $5$ by $14$ .

$\frac{5}{70}$

Step 7.2.17

Cancel the common factor of $5$ and $70$ .

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Step 7.2.17.1

Factor $5$ out of $5$ .

$\frac{5 (1)}{70}$

Step 7.2.17.2

Cancel the common factors.

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Step 7.2.17.2.1

Factor $5$ out of $70$ .

$\frac{5 \cdot 1}{5 \cdot 14}$

Step 7.2.17.2.2

Cancel the common factor.

$\frac{5 \cdot 1}{5 \cdot 14}$

Step 7.2.17.2.3

Rewrite the expression.

$\frac{1}{14}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{14}$

Decimal Form:

$0.0\overline{714285}$

Step 9