Calculus Examples

$\int_{1}^{3} \frac{9}{x^{3}} d x$

Step 1

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$9 \int_{1}^{3} \frac{1}{x^{3}} d x$

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.

$9 \int_{1}^{3} {(x^{3})}^{- 1} d x$

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

$9 \int_{1}^{3} x^{3 \cdot - 1} d x$

Step 2.2.2

Multiply $3$ by $- 1$ .

$9 \int_{1}^{3} x^{- 3} d x$

Step 3

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

$9 (- \frac{1}{2} x^{- 2}]_{1}^{3})$

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 $\frac{1}{2}$ .

$9 (- \frac{x^{- 2}}{2}]_{1}^{3})$

Step 4.1.2

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

$9 (- \frac{1}{2 x^{2}}]_{1}^{3})$

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $- \frac{1}{2 x^{2}}$ at $3$ and at $1$ .

$9 ((- \frac{1}{2 \cdot 3^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Raise $3$ to the power of $2$ .

$9 (- \frac{1}{2 \cdot 9} + \frac{1}{2 \cdot 1^{2}})$

Step 4.2.2.2

Multiply $2$ by $9$ .

$9 (- \frac{1}{18} + \frac{1}{2 \cdot 1^{2}})$

Step 4.2.2.3

One to any power is one.

$9 (- \frac{1}{18} + \frac{1}{2 \cdot 1})$

Step 4.2.2.4

Multiply $2$ by $1$ .

$9 (- \frac{1}{18} + \frac{1}{2})$

Step 4.2.2.5

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$9 (- \frac{1}{18} + \frac{1}{2} \cdot \frac{9}{9})$

Step 4.2.2.6

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.2.6.1

Multiply $\frac{1}{2}$ by $\frac{9}{9}$ .

$9 (- \frac{1}{18} + \frac{9}{2 \cdot 9})$

Step 4.2.2.6.2

Multiply $2$ by $9$ .

$9 (- \frac{1}{18} + \frac{9}{18})$

Step 4.2.2.7

Combine the numerators over the common denominator.

$9 \frac{- 1 + 9}{18}$

Step 4.2.2.8

Add $- 1$ and $9$ .

$9 (\frac{8}{18})$

Step 4.2.2.9

Cancel the common factor of $8$ and $18$ .

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Step 4.2.2.9.1

Factor $2$ out of $8$ .

$9 \frac{2 (4)}{18}$

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 $18$ .

$9 \frac{2 \cdot 4}{2 \cdot 9}$

Step 4.2.2.9.2.2

Cancel the common factor.

$9 \frac{2 \cdot 4}{2 \cdot 9}$

Step 4.2.2.9.2.3

Rewrite the expression.

$9 (\frac{4}{9})$

Step 4.2.2.10

Combine $9$ and $\frac{4}{9}$ .

$\frac{9 \cdot 4}{9}$

Step 4.2.2.11

Multiply $9$ by $4$ .

$\frac{36}{9}$

Step 4.2.2.12

Cancel the common factor of $36$ and $9$ .

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Step 4.2.2.12.1

Factor $9$ out of $36$ .

$\frac{9 \cdot 4}{9}$

Step 4.2.2.12.2

Cancel the common factors.

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Step 4.2.2.12.2.1

Factor $9$ out of $9$ .

$\frac{9 \cdot 4}{9 (1)}$

Step 4.2.2.12.2.2

Cancel the common factor.

$\frac{9 \cdot 4}{9 \cdot 1}$

Step 4.2.2.12.2.3

Rewrite the expression.

$\frac{4}{1}$

Step 4.2.2.12.2.4

Divide $4$ by $1$ .

$4$

Step 5