Calculus Examples

$\int_{1}^{3} \frac{y^{5}}{5} - \frac{1}{12 y^{3}} d y$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{3} \frac{y^{5}}{5} d y + \int_{1}^{3} - \frac{1}{12 y^{3}} d y$

Step 2

Since $\frac{1}{5}$ is constant with respect to $y$ , move $\frac{1}{5}$ out of the integral.

$\frac{1}{5} \int_{1}^{3} y^{5} d y + \int_{1}^{3} - \frac{1}{12 y^{3}} d y$

Step 3

By the Power Rule, the integral of $y^{5}$ with respect to $y$ is $\frac{1}{6} y^{6}$ .

$\frac{1}{5} \frac{1}{6} y^{6}]_{1}^{3} + \int_{1}^{3} - \frac{1}{12 y^{3}} d y$

Step 4

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

$\frac{1}{5} \frac{1}{6} y^{6}]_{1}^{3} - \int_{1}^{3} \frac{1}{12 y^{3}} d y$

Step 5

Since $\frac{1}{12}$ is constant with respect to $y$ , move $\frac{1}{12}$ out of the integral.

$\frac{1}{5} \frac{1}{6} y^{6}]_{1}^{3} - (\frac{1}{12} \int_{1}^{3} \frac{1}{y^{3}} d y)$

Step 6

Apply basic rules of exponents.

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Step 6.1

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

$\frac{1}{5} \frac{1}{6} y^{6}]_{1}^{3} - \frac{1}{12} \int_{1}^{3} {(y^{3})}^{- 1} d y$

Step 6.2

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

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Step 6.2.1

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

$\frac{1}{5} \frac{1}{6} y^{6}]_{1}^{3} - \frac{1}{12} \int_{1}^{3} y^{3 \cdot - 1} d y$

Step 6.2.2

Multiply $3$ by $- 1$ .

$\frac{1}{5} \frac{1}{6} y^{6}]_{1}^{3} - \frac{1}{12} \int_{1}^{3} y^{- 3} d y$

Step 7

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

$\frac{1}{5} \frac{1}{6} y^{6}]_{1}^{3} - \frac{1}{12} - \frac{1}{2} y^{- 2}]_{1}^{3}$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $\frac{1}{6} y^{6}$ at $3$ and at $1$ .

$\frac{1}{5} ((\frac{1}{6} \cdot 3^{6}) - \frac{1}{6} \cdot 1^{6}) - \frac{1}{12} - \frac{1}{2} y^{- 2}]_{1}^{3}$

Step 8.2

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

$\frac{1}{5} (\frac{1}{6} \cdot 3^{6} - \frac{1}{6} \cdot 1^{6}) - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3

Simplify.

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Step 8.3.1

Raise $3$ to the power of $6$ .

$\frac{1}{5} (\frac{1}{6} \cdot 729 - \frac{1}{6} \cdot 1^{6}) - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.2

Combine $\frac{1}{6}$ and $729$ .

$\frac{1}{5} (\frac{729}{6} - \frac{1}{6} \cdot 1^{6}) - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.3

Cancel the common factor of $729$ and $6$ .

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Step 8.3.3.1

Factor $3$ out of $729$ .

$\frac{1}{5} (\frac{3 (243)}{6} - \frac{1}{6} \cdot 1^{6}) - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.3.2

Cancel the common factors.

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Step 8.3.3.2.1

Factor $3$ out of $6$ .

$\frac{1}{5} (\frac{3 \cdot 243}{3 \cdot 2} - \frac{1}{6} \cdot 1^{6}) - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.3.2.2

Cancel the common factor.

$\frac{1}{5} (\frac{3 \cdot 243}{3 \cdot 2} - \frac{1}{6} \cdot 1^{6}) - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.3.2.3

Rewrite the expression.

$\frac{1}{5} (\frac{243}{2} - \frac{1}{6} \cdot 1^{6}) - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.4

One to any power is one.

$\frac{1}{5} (\frac{243}{2} - \frac{1}{6} \cdot 1) - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.5

Multiply $- 1$ by $1$ .

$\frac{1}{5} (\frac{243}{2} - \frac{1}{6}) - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.6

To write $\frac{243}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{5} (\frac{243}{2} \cdot \frac{3}{3} - \frac{1}{6}) - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.7

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.7.1

Multiply $\frac{243}{2}$ by $\frac{3}{3}$ .

$\frac{1}{5} (\frac{243 \cdot 3}{2 \cdot 3} - \frac{1}{6}) - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.7.2

Multiply $2$ by $3$ .

$\frac{1}{5} (\frac{243 \cdot 3}{6} - \frac{1}{6}) - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.8

Combine the numerators over the common denominator.

$\frac{1}{5} \cdot \frac{243 \cdot 3 - 1}{6} - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.9

Simplify the numerator.

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Step 8.3.9.1

Multiply $243$ by $3$ .

$\frac{1}{5} \cdot \frac{729 - 1}{6} - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.9.2

Subtract $1$ from $729$ .

$\frac{1}{5} \cdot \frac{728}{6} - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.10

Cancel the common factor of $728$ and $6$ .

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Step 8.3.10.1

Factor $2$ out of $728$ .

$\frac{1}{5} \cdot \frac{2 (364)}{6} - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.10.2

Cancel the common factors.

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Step 8.3.10.2.1

Factor $2$ out of $6$ .

$\frac{1}{5} \cdot \frac{2 \cdot 364}{2 \cdot 3} - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.10.2.2

Cancel the common factor.

$\frac{1}{5} \cdot \frac{2 \cdot 364}{2 \cdot 3} - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.10.2.3

Rewrite the expression.

$\frac{1}{5} \cdot \frac{364}{3} - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.11

Multiply $\frac{1}{5}$ by $\frac{364}{3}$ .

$\frac{364}{5 \cdot 3} - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.12

Multiply $5$ by $3$ .

$\frac{364}{15} - \frac{1}{12} (- \frac{1}{2} \cdot 3^{- 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.13

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

$\frac{364}{15} - \frac{1}{12} (- \frac{1}{2} \cdot \frac{1}{3^{2}} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.14

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

$\frac{364}{15} - \frac{1}{12} (- \frac{1}{2} \cdot \frac{1}{9} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.15

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

$\frac{364}{15} - \frac{1}{12} (- \frac{1}{9 \cdot 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.16

Multiply $9$ by $2$ .

$\frac{364}{15} - \frac{1}{12} (- \frac{1}{18} + \frac{1}{2} \cdot 1^{- 2})$

Step 8.3.17

One to any power is one.

$\frac{364}{15} - \frac{1}{12} (- \frac{1}{18} + \frac{1}{2} \cdot 1)$

Step 8.3.18

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

$\frac{364}{15} - \frac{1}{12} (- \frac{1}{18} + \frac{1}{2})$

Step 8.3.19

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

$\frac{364}{15} - \frac{1}{12} (- \frac{1}{18} + \frac{1}{2} \cdot \frac{9}{9})$

Step 8.3.20

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

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Step 8.3.20.1

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

$\frac{364}{15} - \frac{1}{12} (- \frac{1}{18} + \frac{9}{2 \cdot 9})$

Step 8.3.20.2

Multiply $2$ by $9$ .

$\frac{364}{15} - \frac{1}{12} (- \frac{1}{18} + \frac{9}{18})$

Step 8.3.21

Combine the numerators over the common denominator.

$\frac{364}{15} - \frac{1}{12} \cdot \frac{- 1 + 9}{18}$

Step 8.3.22

Add $- 1$ and $9$ .

$\frac{364}{15} - \frac{1}{12} \cdot \frac{8}{18}$

Step 8.3.23

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

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Step 8.3.23.1

Factor $2$ out of $8$ .

$\frac{364}{15} - \frac{1}{12} \cdot \frac{2 (4)}{18}$

Step 8.3.23.2

Cancel the common factors.

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Step 8.3.23.2.1

Factor $2$ out of $18$ .

$\frac{364}{15} - \frac{1}{12} \cdot \frac{2 \cdot 4}{2 \cdot 9}$

Step 8.3.23.2.2

Cancel the common factor.

$\frac{364}{15} - \frac{1}{12} \cdot \frac{2 \cdot 4}{2 \cdot 9}$

Step 8.3.23.2.3

Rewrite the expression.

$\frac{364}{15} - \frac{1}{12} \cdot \frac{4}{9}$

Step 8.3.24

Multiply $\frac{4}{9}$ by $\frac{1}{12}$ .

$\frac{364}{15} - \frac{4}{9 \cdot 12}$

Step 8.3.25

Multiply $9$ by $12$ .

$\frac{364}{15} - \frac{4}{108}$

Step 8.3.26

Cancel the common factor of $4$ and $108$ .

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Step 8.3.26.1

Factor $4$ out of $4$ .

$\frac{364}{15} - \frac{4 (1)}{108}$

Step 8.3.26.2

Cancel the common factors.

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Step 8.3.26.2.1

Factor $4$ out of $108$ .

$\frac{364}{15} - \frac{4 \cdot 1}{4 \cdot 27}$

Step 8.3.26.2.2

Cancel the common factor.

$\frac{364}{15} - \frac{4 \cdot 1}{4 \cdot 27}$

Step 8.3.26.2.3

Rewrite the expression.

$\frac{364}{15} - \frac{1}{27}$

Step 8.3.27

To write $\frac{364}{15}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{364}{15} \cdot \frac{9}{9} - \frac{1}{27}$

Step 8.3.28

To write $- \frac{1}{27}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{364}{15} \cdot \frac{9}{9} - \frac{1}{27} \cdot \frac{5}{5}$

Step 8.3.29

Write each expression with a common denominator of $135$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.29.1

Multiply $\frac{364}{15}$ by $\frac{9}{9}$ .

$\frac{364 \cdot 9}{15 \cdot 9} - \frac{1}{27} \cdot \frac{5}{5}$

Step 8.3.29.2

Multiply $15$ by $9$ .

$\frac{364 \cdot 9}{135} - \frac{1}{27} \cdot \frac{5}{5}$

Step 8.3.29.3

Multiply $\frac{1}{27}$ by $\frac{5}{5}$ .

$\frac{364 \cdot 9}{135} - \frac{5}{27 \cdot 5}$

Step 8.3.29.4

Multiply $27$ by $5$ .

$\frac{364 \cdot 9}{135} - \frac{5}{135}$

Step 8.3.30

Combine the numerators over the common denominator.

$\frac{364 \cdot 9 - 5}{135}$

Step 8.3.31

Simplify the numerator.

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Step 8.3.31.1

Multiply $364$ by $9$ .

$\frac{3276 - 5}{135}$

Step 8.3.31.2

Subtract $5$ from $3276$ .

$\frac{3271}{135}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{3271}{135}$

Decimal Form:

$24.2\overline{296}$

Mixed Number Form:

$24 \frac{31}{135}$

Step 10