Calculus Examples

$\int_{1}^{3} \frac{10}{x^{3}} + 3 x d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{3} \frac{10}{x^{3}} d x + \int_{1}^{3} 3 x d x$

Step 2

Since $10$ is constant with respect to $x$ , move $10$ out of the integral.

$10 \int_{1}^{3} \frac{1}{x^{3}} d x + \int_{1}^{3} 3 x d x$

Step 3

Apply basic rules of exponents.

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Step 3.1

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

$10 \int_{1}^{3} {(x^{3})}^{- 1} d x + \int_{1}^{3} 3 x d x$

Step 3.2

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

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Step 3.2.1

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

$10 \int_{1}^{3} x^{3 \cdot - 1} d x + \int_{1}^{3} 3 x d x$

Step 3.2.2

Multiply $3$ by $- 1$ .

$10 \int_{1}^{3} x^{- 3} d x + \int_{1}^{3} 3 x d x$

Step 4

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

$10 (- \frac{1}{2} x^{- 2}]_{1}^{3}) + \int_{1}^{3} 3 x d x$

Step 5

Simplify.

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Step 5.1

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

$10 (- \frac{x^{- 2}}{2}]_{1}^{3}) + \int_{1}^{3} 3 x d x$

Step 5.2

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

$10 (- \frac{1}{2 x^{2}}]_{1}^{3}) + \int_{1}^{3} 3 x d x$

Step 6

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$10 (- \frac{1}{2 x^{2}}]_{1}^{3}) + 3 \int_{1}^{3} x d x$

Step 7

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

$10 (- \frac{1}{2 x^{2}}]_{1}^{3}) + 3 (\frac{1}{2} x^{2}]_{1}^{3})$

Step 8

Simplify the answer.

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Step 8.1

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

$10 (- \frac{1}{2 x^{2}}]_{1}^{3}) + 3 (\frac{x^{2}}{2}]_{1}^{3})$

Step 8.2

Substitute and simplify.

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Step 8.2.1

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

$10 ((- \frac{1}{2 \cdot 3^{2}}) + \frac{1}{2 \cdot 1^{2}}) + 3 (\frac{x^{2}}{2}]_{1}^{3})$

Step 8.2.2

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

$10 ((- \frac{1}{2 \cdot 3^{2}}) + \frac{1}{2 \cdot 1^{2}}) + 3 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 8.2.3

Simplify.

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Step 8.2.3.1

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

$10 (- \frac{1}{2 \cdot 9} + \frac{1}{2 \cdot 1^{2}}) + 3 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 8.2.3.2

Multiply $2$ by $9$ .

$10 (- \frac{1}{18} + \frac{1}{2 \cdot 1^{2}}) + 3 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 8.2.3.3

One to any power is one.

$10 (- \frac{1}{18} + \frac{1}{2 \cdot 1}) + 3 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 8.2.3.4

Multiply $2$ by $1$ .

$10 (- \frac{1}{18} + \frac{1}{2}) + 3 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 8.2.3.5

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

$10 (- \frac{1}{18} + \frac{1}{2} \cdot \frac{9}{9}) + 3 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 8.2.3.6

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

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Step 8.2.3.6.1

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

$10 (- \frac{1}{18} + \frac{9}{2 \cdot 9}) + 3 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 8.2.3.6.2

Multiply $2$ by $9$ .

$10 (- \frac{1}{18} + \frac{9}{18}) + 3 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 8.2.3.7

Combine the numerators over the common denominator.

$10 \frac{- 1 + 9}{18} + 3 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 8.2.3.8

Add $- 1$ and $9$ .

$10 (\frac{8}{18}) + 3 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 8.2.3.9

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

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Step 8.2.3.9.1

Factor $2$ out of $8$ .

$10 \frac{2 (4)}{18} + 3 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 8.2.3.9.2

Cancel the common factors.

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Step 8.2.3.9.2.1

Factor $2$ out of $18$ .

$10 \frac{2 \cdot 4}{2 \cdot 9} + 3 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 8.2.3.9.2.2

Cancel the common factor.

$10 \frac{2 \cdot 4}{2 \cdot 9} + 3 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 8.2.3.9.2.3

Rewrite the expression.

$10 (\frac{4}{9}) + 3 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 8.2.3.10

Combine $10$ and $\frac{4}{9}$ .

$\frac{10 \cdot 4}{9} + 3 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 8.2.3.11

Multiply $10$ by $4$ .

$\frac{40}{9} + 3 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 8.2.3.12

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

$\frac{40}{9} + 3 (\frac{9}{2} - \frac{1^{2}}{2})$

Step 8.2.3.13

One to any power is one.

$\frac{40}{9} + 3 (\frac{9}{2} - \frac{1}{2})$

Step 8.2.3.14

Combine the numerators over the common denominator.

$\frac{40}{9} + 3 \frac{9 - 1}{2}$

Step 8.2.3.15

Subtract $1$ from $9$ .

$\frac{40}{9} + 3 (\frac{8}{2})$

Step 8.2.3.16

Cancel the common factor of $8$ and $2$ .

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Step 8.2.3.16.1

Factor $2$ out of $8$ .

$\frac{40}{9} + 3 \frac{2 \cdot 4}{2}$

Step 8.2.3.16.2

Cancel the common factors.

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Step 8.2.3.16.2.1

Factor $2$ out of $2$ .

$\frac{40}{9} + 3 \frac{2 \cdot 4}{2 (1)}$

Step 8.2.3.16.2.2

Cancel the common factor.

$\frac{40}{9} + 3 \frac{2 \cdot 4}{2 \cdot 1}$

Step 8.2.3.16.2.3

Rewrite the expression.

$\frac{40}{9} + 3 (\frac{4}{1})$

Step 8.2.3.16.2.4

Divide $4$ by $1$ .

$\frac{40}{9} + 3 \cdot 4$

Step 8.2.3.17

Multiply $3$ by $4$ .

$\frac{40}{9} + 12$

Step 8.2.3.18

To write $12$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{40}{9} + 12 \cdot \frac{9}{9}$

Step 8.2.3.19

Combine $12$ and $\frac{9}{9}$ .

$\frac{40}{9} + \frac{12 \cdot 9}{9}$

Step 8.2.3.20

Combine the numerators over the common denominator.

$\frac{40 + 12 \cdot 9}{9}$

Step 8.2.3.21

Simplify the numerator.

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Step 8.2.3.21.1

Multiply $12$ by $9$ .

$\frac{40 + 108}{9}$

Step 8.2.3.21.2

Add $40$ and $108$ .

$\frac{148}{9}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{148}{9}$

Decimal Form:

$16.\overline{4}$

Mixed Number Form:

$16 \frac{4}{9}$

Step 10