Calculus Examples

$\int_{1}^{\sqrt{2}} \frac{u^{5}}{4} - \frac{1}{u^{3}} d u$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{\sqrt{2}} \frac{u^{5}}{4} d u + \int_{1}^{\sqrt{2}} - \frac{1}{u^{3}} d u$

Step 2

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int_{1}^{\sqrt{2}} u^{5} d u + \int_{1}^{\sqrt{2}} - \frac{1}{u^{3}} d u$

Step 3

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

$\frac{1}{4} \frac{1}{6} u^{6}]_{1}^{\sqrt{2}} + \int_{1}^{\sqrt{2}} - \frac{1}{u^{3}} d u$

Step 4

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

$\frac{1}{4} \frac{1}{6} u^{6}]_{1}^{\sqrt{2}} - \int_{1}^{\sqrt{2}} \frac{1}{u^{3}} d u$

Step 5

Apply basic rules of exponents.

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Step 5.1

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

$\frac{1}{4} \frac{1}{6} u^{6}]_{1}^{\sqrt{2}} - \int_{1}^{\sqrt{2}} {(u^{3})}^{- 1} d u$

Step 5.2

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

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Step 5.2.1

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

$\frac{1}{4} \frac{1}{6} u^{6}]_{1}^{\sqrt{2}} - \int_{1}^{\sqrt{2}} u^{3 \cdot - 1} d u$

Step 5.2.2

Multiply $3$ by $- 1$ .

$\frac{1}{4} \frac{1}{6} u^{6}]_{1}^{\sqrt{2}} - \int_{1}^{\sqrt{2}} u^{- 3} d u$

Step 6

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

$\frac{1}{4} \frac{1}{6} u^{6}]_{1}^{\sqrt{2}} - (- \frac{1}{2} u^{- 2}]_{1}^{\sqrt{2}})$

Step 7

Simplify the answer.

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Step 7.1

Simplify.

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Step 7.1.1

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

$\frac{1}{4} \frac{1}{6} u^{6}]_{1}^{\sqrt{2}} - (- \frac{u^{- 2}}{2}]_{1}^{\sqrt{2}})$

Step 7.1.2

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

$\frac{1}{4} \frac{1}{6} u^{6}]_{1}^{\sqrt{2}} - (- \frac{1}{2 u^{2}}]_{1}^{\sqrt{2}})$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{1}{6} u^{6}$ at $\sqrt{2}$ and at $1$ .

$\frac{1}{4} ((\frac{1}{6} {\sqrt{2}}^{6}) - \frac{1}{6} \cdot 1^{6}) - (- \frac{1}{2 u^{2}}]_{1}^{\sqrt{2}})$

Step 7.2.2

Evaluate $- \frac{1}{2 u^{2}}$ at $\sqrt{2}$ and at $1$ .

$\frac{1}{4} ((\frac{1}{6} {\sqrt{2}}^{6}) - \frac{1}{6} \cdot 1^{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Rewrite ${\sqrt{2}}^{6}$ as $2^{3}$ .

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Step 7.2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{1}{4} (\frac{1}{6} {(2^{\frac{1}{2}})}^{6} - \frac{1}{6} \cdot 1^{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.1.2

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

$\frac{1}{4} (\frac{1}{6} \cdot 2^{\frac{1}{2} \cdot 6} - \frac{1}{6} \cdot 1^{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.1.3

Combine $\frac{1}{2}$ and $6$ .

$\frac{1}{4} (\frac{1}{6} \cdot 2^{\frac{6}{2}} - \frac{1}{6} \cdot 1^{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.1.4

Cancel the common factor of $6$ and $2$ .

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Step 7.2.3.1.4.1

Factor $2$ out of $6$ .

$\frac{1}{4} (\frac{1}{6} \cdot 2^{\frac{2 \cdot 3}{2}} - \frac{1}{6} \cdot 1^{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.1.4.2

Cancel the common factors.

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Step 7.2.3.1.4.2.1

Factor $2$ out of $2$ .

$\frac{1}{4} (\frac{1}{6} \cdot 2^{\frac{2 \cdot 3}{2 (1)}} - \frac{1}{6} \cdot 1^{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.1.4.2.2

Cancel the common factor.

$\frac{1}{4} (\frac{1}{6} \cdot 2^{\frac{2 \cdot 3}{2 \cdot 1}} - \frac{1}{6} \cdot 1^{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.1.4.2.3

Rewrite the expression.

$\frac{1}{4} (\frac{1}{6} \cdot 2^{\frac{3}{1}} - \frac{1}{6} \cdot 1^{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.1.4.2.4

Divide $3$ by $1$ .

$\frac{1}{4} (\frac{1}{6} \cdot 2^{3} - \frac{1}{6} \cdot 1^{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.2

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

$\frac{1}{4} (\frac{1}{6} \cdot 8 - \frac{1}{6} \cdot 1^{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.3

Combine $\frac{1}{6}$ and $8$ .

$\frac{1}{4} (\frac{8}{6} - \frac{1}{6} \cdot 1^{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.4

Cancel the common factor of $8$ and $6$ .

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Step 7.2.3.4.1

Factor $2$ out of $8$ .

$\frac{1}{4} (\frac{2 (4)}{6} - \frac{1}{6} \cdot 1^{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.4.2

Cancel the common factors.

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Step 7.2.3.4.2.1

Factor $2$ out of $6$ .

$\frac{1}{4} (\frac{2 \cdot 4}{2 \cdot 3} - \frac{1}{6} \cdot 1^{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.4.2.2

Cancel the common factor.

$\frac{1}{4} (\frac{2 \cdot 4}{2 \cdot 3} - \frac{1}{6} \cdot 1^{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.4.2.3

Rewrite the expression.

$\frac{1}{4} (\frac{4}{3} - \frac{1}{6} \cdot 1^{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.5

One to any power is one.

$\frac{1}{4} (\frac{4}{3} - \frac{1}{6} \cdot 1) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.6

Multiply $- 1$ by $1$ .

$\frac{1}{4} (\frac{4}{3} - \frac{1}{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.7

To write $\frac{4}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{4} (\frac{4}{3} \cdot \frac{2}{2} - \frac{1}{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.8

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

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Step 7.2.3.8.1

Multiply $\frac{4}{3}$ by $\frac{2}{2}$ .

$\frac{1}{4} (\frac{4 \cdot 2}{3 \cdot 2} - \frac{1}{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.8.2

Multiply $3$ by $2$ .

$\frac{1}{4} (\frac{4 \cdot 2}{6} - \frac{1}{6}) - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.9

Combine the numerators over the common denominator.

$\frac{1}{4} \cdot \frac{4 \cdot 2 - 1}{6} - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.10

Simplify the numerator.

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Step 7.2.3.10.1

Multiply $4$ by $2$ .

$\frac{1}{4} \cdot \frac{8 - 1}{6} - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.10.2

Subtract $1$ from $8$ .

$\frac{1}{4} \cdot \frac{7}{6} - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.11

Multiply $\frac{1}{4}$ by $\frac{7}{6}$ .

$\frac{7}{4 \cdot 6} - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.12

Multiply $4$ by $6$ .

$\frac{7}{24} - ((- \frac{1}{2 {\sqrt{2}}^{2}}) + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.13

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 7.2.3.13.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{7}{24} - (- \frac{1}{2 {(2^{\frac{1}{2}})}^{2}} + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.13.2

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

$\frac{7}{24} - (- \frac{1}{2 \cdot 2^{\frac{1}{2} \cdot 2}} + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.13.3

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

$\frac{7}{24} - (- \frac{1}{2 \cdot 2^{\frac{2}{2}}} + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.13.4

Cancel the common factor of $2$ .

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Step 7.2.3.13.4.1

Cancel the common factor.

$\frac{7}{24} - (- \frac{1}{2 \cdot 2^{\frac{2}{2}}} + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.13.4.2

Rewrite the expression.

$\frac{7}{24} - (- \frac{1}{2 \cdot 2^{1}} + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.13.5

Evaluate the exponent.

$\frac{7}{24} - (- \frac{1}{2 \cdot 2} + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.14

Multiply $2$ by $2$ .

$\frac{7}{24} - (- \frac{1}{4} + \frac{1}{2 \cdot 1^{2}})$

Step 7.2.3.15

One to any power is one.

$\frac{7}{24} - (- \frac{1}{4} + \frac{1}{2 \cdot 1})$

Step 7.2.3.16

Multiply $2$ by $1$ .

$\frac{7}{24} - (- \frac{1}{4} + \frac{1}{2})$

Step 7.2.3.17

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

$\frac{7}{24} - (- \frac{1}{4} + \frac{1}{2} \cdot \frac{2}{2})$

Step 7.2.3.18

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.3.18.1

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

$\frac{7}{24} - (- \frac{1}{4} + \frac{2}{2 \cdot 2})$

Step 7.2.3.18.2

Multiply $2$ by $2$ .

$\frac{7}{24} - (- \frac{1}{4} + \frac{2}{4})$

Step 7.2.3.19

Combine the numerators over the common denominator.

$\frac{7}{24} - \frac{- 1 + 2}{4}$

Step 7.2.3.20

Add $- 1$ and $2$ .

$\frac{7}{24} - \frac{1}{4}$

Step 7.2.3.21

To write $- \frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{7}{24} - \frac{1}{4} \cdot \frac{6}{6}$

Step 7.2.3.22

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.3.22.1

Multiply $\frac{1}{4}$ by $\frac{6}{6}$ .

$\frac{7}{24} - \frac{6}{4 \cdot 6}$

Step 7.2.3.22.2

Multiply $4$ by $6$ .

$\frac{7}{24} - \frac{6}{24}$

Step 7.2.3.23

Combine the numerators over the common denominator.

$\frac{7 - 6}{24}$

Step 7.2.3.24

Subtract $6$ from $7$ .

$\frac{1}{24}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{24}$

Decimal Form:

$0.041\overline{6}$

Step 9