Calculus Examples

$\int_{1}^{3} 5 n^{- 2} - n^{- 3} d n$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{3} 5 n^{- 2} d n + \int_{1}^{3} - n^{- 3} d n$

Step 2

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

$5 \int_{1}^{3} n^{- 2} d n + \int_{1}^{3} - n^{- 3} d n$

Step 3

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

$5 (- n^{- 1}]_{1}^{3}) + \int_{1}^{3} - n^{- 3} d n$

Step 4

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

$5 (- n^{- 1}]_{1}^{3}) - \int_{1}^{3} n^{- 3} d n$

Step 5

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

$5 (- n^{- 1}]_{1}^{3}) - (- \frac{1}{2} n^{- 2}]_{1}^{3})$

Step 6

Simplify the answer.

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Step 6.1

Simplify.

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Step 6.1.1

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

$5 (- n^{- 1}]_{1}^{3}) - (- \frac{n^{- 2}}{2}]_{1}^{3})$

Step 6.1.2

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

$5 (- n^{- 1}]_{1}^{3}) - (- \frac{1}{2 n^{2}}]_{1}^{3})$

Step 6.2

Substitute and simplify.

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Step 6.2.1

Evaluate $- n^{- 1}$ at $3$ and at $1$ .

$5 ((- 3^{- 1}) + 1^{- 1}) - (- \frac{1}{2 n^{2}}]_{1}^{3})$

Step 6.2.2

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

$5 ((- 3^{- 1}) + 1^{- 1}) - ((- \frac{1}{2 \cdot 3^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 6.2.3

Simplify.

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Step 6.2.3.1

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

$5 (- \frac{1}{3} + 1^{- 1}) - ((- \frac{1}{2 \cdot 3^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 6.2.3.2

One to any power is one.

$5 (- \frac{1}{3} + 1) - ((- \frac{1}{2 \cdot 3^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 6.2.3.3

Write $1$ as a fraction with a common denominator.

$5 (- \frac{1}{3} + \frac{3}{3}) - ((- \frac{1}{2 \cdot 3^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 6.2.3.4

Combine the numerators over the common denominator.

$5 \frac{- 1 + 3}{3} - ((- \frac{1}{2 \cdot 3^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 6.2.3.5

Add $- 1$ and $3$ .

$5 (\frac{2}{3}) - ((- \frac{1}{2 \cdot 3^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 6.2.3.6

Combine $5$ and $\frac{2}{3}$ .

$\frac{5 \cdot 2}{3} - ((- \frac{1}{2 \cdot 3^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 6.2.3.7

Multiply $5$ by $2$ .

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

Step 6.2.3.8

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

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

Step 6.2.3.9

Multiply $2$ by $9$ .

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

Step 6.2.3.10

One to any power is one.

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

Step 6.2.3.11

Multiply $2$ by $1$ .

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

Step 6.2.3.12

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

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

Step 6.2.3.13

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

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Step 6.2.3.13.1

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

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

Step 6.2.3.13.2

Multiply $2$ by $9$ .

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

Step 6.2.3.14

Combine the numerators over the common denominator.

$\frac{10}{3} - \frac{- 1 + 9}{18}$

Step 6.2.3.15

Add $- 1$ and $9$ .

$\frac{10}{3} - \frac{8}{18}$

Step 6.2.3.16

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

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Step 6.2.3.16.1

Factor $2$ out of $8$ .

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

Step 6.2.3.16.2

Cancel the common factors.

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Step 6.2.3.16.2.1

Factor $2$ out of $18$ .

$\frac{10}{3} - \frac{2 \cdot 4}{2 \cdot 9}$

Step 6.2.3.16.2.2

Cancel the common factor.

$\frac{10}{3} - \frac{2 \cdot 4}{2 \cdot 9}$

Step 6.2.3.16.2.3

Rewrite the expression.

$\frac{10}{3} - \frac{4}{9}$

Step 6.2.3.17

To write $\frac{10}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{10}{3} \cdot \frac{3}{3} - \frac{4}{9}$

Step 6.2.3.18

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.3.18.1

Multiply $\frac{10}{3}$ by $\frac{3}{3}$ .

$\frac{10 \cdot 3}{3 \cdot 3} - \frac{4}{9}$

Step 6.2.3.18.2

Multiply $3$ by $3$ .

$\frac{10 \cdot 3}{9} - \frac{4}{9}$

Step 6.2.3.19

Combine the numerators over the common denominator.

$\frac{10 \cdot 3 - 4}{9}$

Step 6.2.3.20

Simplify the numerator.

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Step 6.2.3.20.1

Multiply $10$ by $3$ .

$\frac{30 - 4}{9}$

Step 6.2.3.20.2

Subtract $4$ from $30$ .

$\frac{26}{9}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{26}{9}$

Decimal Form:

$2.\overline{8}$

Mixed Number Form:

$2 \frac{8}{9}$

Step 8