Calculus Examples

$\int_{5}^{7} - \frac{2}{x^{3}} d x$

Step 1

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

$- \int_{5}^{7} \frac{2}{x^{3}} d x$

Step 2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$- (2 \int_{5}^{7} \frac{1}{x^{3}} d x)$

Step 3

Simplify the expression.

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Step 3.1

Multiply $2$ by $- 1$ .

$- 2 \int_{5}^{7} \frac{1}{x^{3}} d x$

Step 3.2

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$- 2 \int_{5}^{7} {(x^{3})}^{- 1} d x$

Step 3.3

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 3.3.1

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

$- 2 \int_{5}^{7} x^{3 \cdot - 1} d x$

Step 3.3.2

Multiply $3$ by $- 1$ .

$- 2 \int_{5}^{7} x^{- 3} d x$

Step 4

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

$- 2 (- \frac{1}{2} x^{- 2}]_{5}^{7})$

Step 5

Simplify the answer.

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Step 5.1

Simplify.

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Step 5.1.1

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

$- 2 (- \frac{x^{- 2}}{2}]_{5}^{7})$

Step 5.1.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 2 (- \frac{1}{2 x^{2}}]_{5}^{7})$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $- \frac{1}{2 x^{2}}$ at $7$ and at $5$ .

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

Raise $7$ to the power of $2$ .

$- 2 (- \frac{1}{2 \cdot 49} + \frac{1}{2 \cdot 5^{2}})$

Step 5.2.2.2

Multiply $2$ by $49$ .

$- 2 (- \frac{1}{98} + \frac{1}{2 \cdot 5^{2}})$

Step 5.2.2.3

Raise $5$ to the power of $2$ .

$- 2 (- \frac{1}{98} + \frac{1}{2 \cdot 25})$

Step 5.2.2.4

Multiply $2$ by $25$ .

$- 2 (- \frac{1}{98} + \frac{1}{50})$

Step 5.2.2.5

To write $- \frac{1}{98}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$- 2 (- \frac{1}{98} \cdot \frac{25}{25} + \frac{1}{50})$

Step 5.2.2.6

To write $\frac{1}{50}$ as a fraction with a common denominator, multiply by $\frac{49}{49}$ .

$- 2 (- \frac{1}{98} \cdot \frac{25}{25} + \frac{1}{50} \cdot \frac{49}{49})$

Step 5.2.2.7

Write each expression with a common denominator of $2450$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.2.7.1

Multiply $\frac{1}{98}$ by $\frac{25}{25}$ .

$- 2 (- \frac{25}{98 \cdot 25} + \frac{1}{50} \cdot \frac{49}{49})$

Step 5.2.2.7.2

Multiply $98$ by $25$ .

$- 2 (- \frac{25}{2450} + \frac{1}{50} \cdot \frac{49}{49})$

Step 5.2.2.7.3

Multiply $\frac{1}{50}$ by $\frac{49}{49}$ .

$- 2 (- \frac{25}{2450} + \frac{49}{50 \cdot 49})$

Step 5.2.2.7.4

Multiply $50$ by $49$ .

$- 2 (- \frac{25}{2450} + \frac{49}{2450})$

Step 5.2.2.8

Combine the numerators over the common denominator.

$- 2 \frac{- 25 + 49}{2450}$

Step 5.2.2.9

Add $- 25$ and $49$ .

$- 2 (\frac{24}{2450})$

Step 5.2.2.10

Cancel the common factor of $24$ and $2450$ .

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Step 5.2.2.10.1

Factor $2$ out of $24$ .

$- 2 \frac{2 (12)}{2450}$

Step 5.2.2.10.2

Cancel the common factors.

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Step 5.2.2.10.2.1

Factor $2$ out of $2450$ .

$- 2 \frac{2 \cdot 12}{2 \cdot 1225}$

Step 5.2.2.10.2.2

Cancel the common factor.

$- 2 \frac{2 \cdot 12}{2 \cdot 1225}$

Step 5.2.2.10.2.3

Rewrite the expression.

$- 2 (\frac{12}{1225})$

Step 5.2.2.11

Combine $- 2$ and $\frac{12}{1225}$ .

$\frac{- 2 \cdot 12}{1225}$

Step 5.2.2.12

Multiply $- 2$ by $12$ .

$\frac{- 24}{1225}$

Step 5.2.2.13

Move the negative in front of the fraction.

$- \frac{24}{1225}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{24}{1225}$

Decimal Form:

$- 0.01959183 \dots$

Step 7