Calculus Examples

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

Step 1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

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

Step 2

Apply basic rules of exponents.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply $3$ by $- 1$ .

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

Simplify.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.2

Substitute and simplify.

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Step 4.2.1

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

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

$5 (- \frac{1}{2 \cdot 25} + \frac{1}{2 \cdot 1^{2}})$

Step 4.2.2.2

Multiply $2$ by $25$ .

$5 (- \frac{1}{50} + \frac{1}{2 \cdot 1^{2}})$

Step 4.2.2.3

One to any power is one.

$5 (- \frac{1}{50} + \frac{1}{2 \cdot 1})$

Step 4.2.2.4

Multiply $2$ by $1$ .

$5 (- \frac{1}{50} + \frac{1}{2})$

Step 4.2.2.5

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$5 (- \frac{1}{50} + \frac{1}{2} \cdot \frac{25}{25})$

Step 4.2.2.6

Write each expression with a common denominator of $50$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.2.6.1

Multiply $\frac{1}{2}$ by $\frac{25}{25}$ .

$5 (- \frac{1}{50} + \frac{25}{2 \cdot 25})$

Step 4.2.2.6.2

Multiply $2$ by $25$ .

$5 (- \frac{1}{50} + \frac{25}{50})$

Step 4.2.2.7

Combine the numerators over the common denominator.

$5 \frac{- 1 + 25}{50}$

Step 4.2.2.8

Add $- 1$ and $25$ .

$5 (\frac{24}{50})$

Step 4.2.2.9

Cancel the common factor of $24$ and $50$ .

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Step 4.2.2.9.1

Factor $2$ out of $24$ .

$5 \frac{2 (12)}{50}$

Step 4.2.2.9.2

Cancel the common factors.

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Step 4.2.2.9.2.1

Factor $2$ out of $50$ .

$5 \frac{2 \cdot 12}{2 \cdot 25}$

Step 4.2.2.9.2.2

Cancel the common factor.

$5 \frac{2 \cdot 12}{2 \cdot 25}$

Step 4.2.2.9.2.3

Rewrite the expression.

$5 (\frac{12}{25})$

Step 4.2.2.10

Combine $5$ and $\frac{12}{25}$ .

$\frac{5 \cdot 12}{25}$

Step 4.2.2.11

Multiply $5$ by $12$ .

$\frac{60}{25}$

Step 4.2.2.12

Cancel the common factor of $60$ and $25$ .

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Step 4.2.2.12.1

Factor $5$ out of $60$ .

$\frac{5 (12)}{25}$

Step 4.2.2.12.2

Cancel the common factors.

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Step 4.2.2.12.2.1

Factor $5$ out of $25$ .

$\frac{5 \cdot 12}{5 \cdot 5}$

Step 4.2.2.12.2.2

Cancel the common factor.

$\frac{5 \cdot 12}{5 \cdot 5}$

Step 4.2.2.12.2.3

Rewrite the expression.

$\frac{12}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{12}{5}$

Decimal Form:

$2.4$

Mixed Number Form:

$2 \frac{2}{5}$

Step 6