Calculus Examples

$\int_{0}^{1} 3 + \frac{1}{4} u^{4} - \frac{2}{3} u^{9} d u$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{1} 3 d u + \int_{0}^{1} \frac{1}{4} u^{4} d u + \int_{0}^{1} - \frac{2}{3} u^{9} d u$

Step 2

Apply the constant rule.

$3 u]_{0}^{1} + \int_{0}^{1} \frac{1}{4} u^{4} d u + \int_{0}^{1} - \frac{2}{3} u^{9} d u$

Step 3

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

$3 u]_{0}^{1} + \frac{1}{4} \int_{0}^{1} u^{4} d u + \int_{0}^{1} - \frac{2}{3} u^{9} d u$

Step 4

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$3 u]_{0}^{1} + \frac{1}{4} \frac{1}{5} u^{5}]_{0}^{1} + \int_{0}^{1} - \frac{2}{3} u^{9} d u$

Step 5

Since $- \frac{2}{3}$ is constant with respect to $u$ , move $- \frac{2}{3}$ out of the integral.

$3 u]_{0}^{1} + \frac{1}{4} \frac{1}{5} u^{5}]_{0}^{1} - \frac{2}{3} \int_{0}^{1} u^{9} d u$

Step 6

By the Power Rule, the integral of $u^{9}$ with respect to $u$ is $\frac{1}{10} u^{10}$ .

$3 u]_{0}^{1} + \frac{1}{4} \frac{1}{5} u^{5}]_{0}^{1} - \frac{2}{3} \frac{1}{10} u^{10}]_{0}^{1}$

Step 7

Substitute and simplify.

Tap for more steps...

Step 7.1

Evaluate $3 u$ at $1$ and at $0$ .

$(3 \cdot 1) - 3 \cdot 0 + \frac{1}{4} \frac{1}{5} u^{5}]_{0}^{1} - \frac{2}{3} \frac{1}{10} u^{10}]_{0}^{1}$

Step 7.2

Evaluate $\frac{1}{5} u^{5}$ at $1$ and at $0$ .

$3 \cdot 1 - 3 \cdot 0 + \frac{1}{4} (\frac{1}{5} \cdot 1^{5} - \frac{1}{5} \cdot 0^{5}) - \frac{2}{3} \frac{1}{10} u^{10}]_{0}^{1}$

Step 7.3

Evaluate $\frac{1}{10} u^{10}$ at $1$ and at $0$ .

$3 \cdot 1 - 3 \cdot 0 + \frac{1}{4} (\frac{1}{5} \cdot 1^{5} - \frac{1}{5} \cdot 0^{5}) - \frac{2}{3} ((\frac{1}{10} \cdot 1^{10}) - \frac{1}{10} \cdot 0^{10})$

Step 7.4

Simplify.

Tap for more steps...

Step 7.4.1

Multiply $3$ by $1$ .

$3 - 3 \cdot 0 + \frac{1}{4} (\frac{1}{5} \cdot 1^{5} - \frac{1}{5} \cdot 0^{5}) - \frac{2}{3} ((\frac{1}{10} \cdot 1^{10}) - \frac{1}{10} \cdot 0^{10})$

Step 7.4.2

Multiply $- 3$ by $0$ .

$3 + 0 + \frac{1}{4} (\frac{1}{5} \cdot 1^{5} - \frac{1}{5} \cdot 0^{5}) - \frac{2}{3} ((\frac{1}{10} \cdot 1^{10}) - \frac{1}{10} \cdot 0^{10})$

Step 7.4.3

Add $3$ and $0$ .

$3 + \frac{1}{4} (\frac{1}{5} \cdot 1^{5} - \frac{1}{5} \cdot 0^{5}) - \frac{2}{3} ((\frac{1}{10} \cdot 1^{10}) - \frac{1}{10} \cdot 0^{10})$

Step 7.4.4

One to any power is one.

$3 + \frac{1}{4} (\frac{1}{5} \cdot 1 - \frac{1}{5} \cdot 0^{5}) - \frac{2}{3} ((\frac{1}{10} \cdot 1^{10}) - \frac{1}{10} \cdot 0^{10})$

Step 7.4.5

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

$3 + \frac{1}{4} (\frac{1}{5} - \frac{1}{5} \cdot 0^{5}) - \frac{2}{3} ((\frac{1}{10} \cdot 1^{10}) - \frac{1}{10} \cdot 0^{10})$

Step 7.4.6

Raising $0$ to any positive power yields $0$ .

$3 + \frac{1}{4} (\frac{1}{5} - \frac{1}{5} \cdot 0) - \frac{2}{3} ((\frac{1}{10} \cdot 1^{10}) - \frac{1}{10} \cdot 0^{10})$

Step 7.4.7

Multiply $0$ by $- 1$ .

$3 + \frac{1}{4} (\frac{1}{5} + 0 (\frac{1}{5})) - \frac{2}{3} ((\frac{1}{10} \cdot 1^{10}) - \frac{1}{10} \cdot 0^{10})$

Step 7.4.8

Multiply $0$ by $\frac{1}{5}$ .

$3 + \frac{1}{4} (\frac{1}{5} + 0) - \frac{2}{3} ((\frac{1}{10} \cdot 1^{10}) - \frac{1}{10} \cdot 0^{10})$

Step 7.4.9

Add $\frac{1}{5}$ and $0$ .

$3 + \frac{1}{4} \cdot \frac{1}{5} - \frac{2}{3} ((\frac{1}{10} \cdot 1^{10}) - \frac{1}{10} \cdot 0^{10})$

Step 7.4.10

Multiply $\frac{1}{4}$ by $\frac{1}{5}$ .

$3 + \frac{1}{4 \cdot 5} - \frac{2}{3} ((\frac{1}{10} \cdot 1^{10}) - \frac{1}{10} \cdot 0^{10})$

Step 7.4.11

Multiply $4$ by $5$ .

$3 + \frac{1}{20} - \frac{2}{3} ((\frac{1}{10} \cdot 1^{10}) - \frac{1}{10} \cdot 0^{10})$

Step 7.4.12

To write $3$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$3 \cdot \frac{20}{20} + \frac{1}{20} - \frac{2}{3} ((\frac{1}{10} \cdot 1^{10}) - \frac{1}{10} \cdot 0^{10})$

Step 7.4.13

Combine $3$ and $\frac{20}{20}$ .

$\frac{3 \cdot 20}{20} + \frac{1}{20} - \frac{2}{3} ((\frac{1}{10} \cdot 1^{10}) - \frac{1}{10} \cdot 0^{10})$

Step 7.4.14

Combine the numerators over the common denominator.

$\frac{3 \cdot 20 + 1}{20} - \frac{2}{3} ((\frac{1}{10} \cdot 1^{10}) - \frac{1}{10} \cdot 0^{10})$

Step 7.4.15

Simplify the numerator.

Tap for more steps...

Step 7.4.15.1

Multiply $3$ by $20$ .

$\frac{60 + 1}{20} - \frac{2}{3} ((\frac{1}{10} \cdot 1^{10}) - \frac{1}{10} \cdot 0^{10})$

Step 7.4.15.2

Add $60$ and $1$ .

$\frac{61}{20} - \frac{2}{3} ((\frac{1}{10} \cdot 1^{10}) - \frac{1}{10} \cdot 0^{10})$

Step 7.4.16

One to any power is one.

$\frac{61}{20} - \frac{2}{3} (\frac{1}{10} \cdot 1 - \frac{1}{10} \cdot 0^{10})$

Step 7.4.17

Multiply $\frac{1}{10}$ by $1$ .

$\frac{61}{20} - \frac{2}{3} (\frac{1}{10} - \frac{1}{10} \cdot 0^{10})$

Step 7.4.18

Raising $0$ to any positive power yields $0$ .

$\frac{61}{20} - \frac{2}{3} (\frac{1}{10} - \frac{1}{10} \cdot 0)$

Step 7.4.19

Multiply $0$ by $- 1$ .

$\frac{61}{20} - \frac{2}{3} (\frac{1}{10} + 0 (\frac{1}{10}))$

Step 7.4.20

Multiply $0$ by $\frac{1}{10}$ .

$\frac{61}{20} - \frac{2}{3} (\frac{1}{10} + 0)$

Step 7.4.21

Add $\frac{1}{10}$ and $0$ .

$\frac{61}{20} - \frac{2}{3} \cdot \frac{1}{10}$

Step 7.4.22

Multiply $\frac{1}{10}$ by $\frac{2}{3}$ .

$\frac{61}{20} - \frac{2}{10 \cdot 3}$

Step 7.4.23

Multiply $10$ by $3$ .

$\frac{61}{20} - \frac{2}{30}$

Step 7.4.24

Cancel the common factor of $2$ and $30$ .

Tap for more steps...

Step 7.4.24.1

Factor $2$ out of $2$ .

$\frac{61}{20} - \frac{2 (1)}{30}$

Step 7.4.24.2

Cancel the common factors.

Tap for more steps...

Step 7.4.24.2.1

Factor $2$ out of $30$ .

$\frac{61}{20} - \frac{2 \cdot 1}{2 \cdot 15}$

Step 7.4.24.2.2

Cancel the common factor.

$\frac{61}{20} - \frac{2 \cdot 1}{2 \cdot 15}$

Step 7.4.24.2.3

Rewrite the expression.

$\frac{61}{20} - \frac{1}{15}$

Step 7.4.25

To write $\frac{61}{20}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{61}{20} \cdot \frac{3}{3} - \frac{1}{15}$

Step 7.4.26

To write $- \frac{1}{15}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{61}{20} \cdot \frac{3}{3} - \frac{1}{15} \cdot \frac{4}{4}$

Step 7.4.27

Write each expression with a common denominator of $60$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 7.4.27.1

Multiply $\frac{61}{20}$ by $\frac{3}{3}$ .

$\frac{61 \cdot 3}{20 \cdot 3} - \frac{1}{15} \cdot \frac{4}{4}$

Step 7.4.27.2

Multiply $20$ by $3$ .

$\frac{61 \cdot 3}{60} - \frac{1}{15} \cdot \frac{4}{4}$

Step 7.4.27.3

Multiply $\frac{1}{15}$ by $\frac{4}{4}$ .

$\frac{61 \cdot 3}{60} - \frac{4}{15 \cdot 4}$

Step 7.4.27.4

Multiply $15$ by $4$ .

$\frac{61 \cdot 3}{60} - \frac{4}{60}$

Step 7.4.28

Combine the numerators over the common denominator.

$\frac{61 \cdot 3 - 4}{60}$

Step 7.4.29

Simplify the numerator.

Tap for more steps...

Step 7.4.29.1

Multiply $61$ by $3$ .

$\frac{183 - 4}{60}$

Step 7.4.29.2

Subtract $4$ from $183$ .

$\frac{179}{60}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{179}{60}$

Decimal Form:

$2.98\overline{3}$

Mixed Number Form:

$2 \frac{59}{60}$

Step 9