Calculus Examples

$\int_{0}^{1} \frac{2}{7} x^{3} + \frac{1}{5} x^{2} + \frac{1}{2} x d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{1} \frac{2}{7} x^{3} d x + \int_{0}^{1} \frac{1}{5} x^{2} d x + \int_{0}^{1} \frac{1}{2} x d x$

Step 2

Since $\frac{2}{7}$ is constant with respect to $x$ , move $\frac{2}{7}$ out of the integral.

$\frac{2}{7} \int_{0}^{1} x^{3} d x + \int_{0}^{1} \frac{1}{5} x^{2} d x + \int_{0}^{1} \frac{1}{2} x d x$

Step 3

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$\frac{2}{7} \frac{1}{4} x^{4}]_{0}^{1} + \int_{0}^{1} \frac{1}{5} x^{2} d x + \int_{0}^{1} \frac{1}{2} x d x$

Step 4

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

$\frac{2}{7} \frac{1}{4} x^{4}]_{0}^{1} + \frac{1}{5} \int_{0}^{1} x^{2} d x + \int_{0}^{1} \frac{1}{2} x d x$

Step 5

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

$\frac{2}{7} \frac{1}{4} x^{4}]_{0}^{1} + \frac{1}{5} \frac{1}{3} x^{3}]_{0}^{1} + \int_{0}^{1} \frac{1}{2} x d x$

Step 6

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{2}{7} \frac{1}{4} x^{4}]_{0}^{1} + \frac{1}{5} \frac{1}{3} x^{3}]_{0}^{1} + \frac{1}{2} \int_{0}^{1} x d x$

Step 7

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

$\frac{2}{7} \frac{1}{4} x^{4}]_{0}^{1} + \frac{1}{5} \frac{1}{3} x^{3}]_{0}^{1} + \frac{1}{2} \frac{1}{2} x^{2}]_{0}^{1}$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $\frac{1}{4} x^{4}$ at $1$ and at $0$ .

$\frac{2}{7} ((\frac{1}{4} \cdot 1^{4}) - \frac{1}{4} \cdot 0^{4}) + \frac{1}{5} \frac{1}{3} x^{3}]_{0}^{1} + \frac{1}{2} \frac{1}{2} x^{2}]_{0}^{1}$

Step 8.2

Evaluate $\frac{1}{3} x^{3}$ at $1$ and at $0$ .

$\frac{2}{7} (\frac{1}{4} \cdot 1^{4} - \frac{1}{4} \cdot 0^{4}) + \frac{1}{5} (\frac{1}{3} \cdot 1^{3} - \frac{1}{3} \cdot 0^{3}) + \frac{1}{2} \frac{1}{2} x^{2}]_{0}^{1}$

Step 8.3

Evaluate $\frac{1}{2} x^{2}$ at $1$ and at $0$ .

$\frac{2}{7} (\frac{1}{4} \cdot 1^{4} - \frac{1}{4} \cdot 0^{4}) + \frac{1}{5} (\frac{1}{3} \cdot 1^{3} - \frac{1}{3} \cdot 0^{3}) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4

Simplify.

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Step 8.4.1

One to any power is one.

$\frac{2}{7} (\frac{1}{4} \cdot 1 - \frac{1}{4} \cdot 0^{4}) + \frac{1}{5} (\frac{1}{3} \cdot 1^{3} - \frac{1}{3} \cdot 0^{3}) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.2

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

$\frac{2}{7} (\frac{1}{4} - \frac{1}{4} \cdot 0^{4}) + \frac{1}{5} (\frac{1}{3} \cdot 1^{3} - \frac{1}{3} \cdot 0^{3}) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.3

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

$\frac{2}{7} (\frac{1}{4} - \frac{1}{4} \cdot 0) + \frac{1}{5} (\frac{1}{3} \cdot 1^{3} - \frac{1}{3} \cdot 0^{3}) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.4

Multiply $0$ by $- 1$ .

$\frac{2}{7} (\frac{1}{4} + 0 (\frac{1}{4})) + \frac{1}{5} (\frac{1}{3} \cdot 1^{3} - \frac{1}{3} \cdot 0^{3}) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.5

Multiply $0$ by $\frac{1}{4}$ .

$\frac{2}{7} (\frac{1}{4} + 0) + \frac{1}{5} (\frac{1}{3} \cdot 1^{3} - \frac{1}{3} \cdot 0^{3}) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.6

Add $\frac{1}{4}$ and $0$ .

$\frac{2}{7} \cdot \frac{1}{4} + \frac{1}{5} (\frac{1}{3} \cdot 1^{3} - \frac{1}{3} \cdot 0^{3}) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.7

Multiply $\frac{2}{7}$ by $\frac{1}{4}$ .

$\frac{2}{7 \cdot 4} + \frac{1}{5} (\frac{1}{3} \cdot 1^{3} - \frac{1}{3} \cdot 0^{3}) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.8

Multiply $7$ by $4$ .

$\frac{2}{28} + \frac{1}{5} (\frac{1}{3} \cdot 1^{3} - \frac{1}{3} \cdot 0^{3}) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.9

Cancel the common factor of $2$ and $28$ .

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Step 8.4.9.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{28} + \frac{1}{5} (\frac{1}{3} \cdot 1^{3} - \frac{1}{3} \cdot 0^{3}) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.9.2

Cancel the common factors.

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Step 8.4.9.2.1

Factor $2$ out of $28$ .

$\frac{2 \cdot 1}{2 \cdot 14} + \frac{1}{5} (\frac{1}{3} \cdot 1^{3} - \frac{1}{3} \cdot 0^{3}) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.9.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 14} + \frac{1}{5} (\frac{1}{3} \cdot 1^{3} - \frac{1}{3} \cdot 0^{3}) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.9.2.3

Rewrite the expression.

$\frac{1}{14} + \frac{1}{5} (\frac{1}{3} \cdot 1^{3} - \frac{1}{3} \cdot 0^{3}) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.10

One to any power is one.

$\frac{1}{14} + \frac{1}{5} (\frac{1}{3} \cdot 1 - \frac{1}{3} \cdot 0^{3}) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.11

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

$\frac{1}{14} + \frac{1}{5} (\frac{1}{3} - \frac{1}{3} \cdot 0^{3}) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.12

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

$\frac{1}{14} + \frac{1}{5} (\frac{1}{3} - \frac{1}{3} \cdot 0) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.13

Multiply $0$ by $- 1$ .

$\frac{1}{14} + \frac{1}{5} (\frac{1}{3} + 0 (\frac{1}{3})) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.14

Multiply $0$ by $\frac{1}{3}$ .

$\frac{1}{14} + \frac{1}{5} (\frac{1}{3} + 0) + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.15

Add $\frac{1}{3}$ and $0$ .

$\frac{1}{14} + \frac{1}{5} \cdot \frac{1}{3} + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.16

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

$\frac{1}{14} + \frac{1}{5 \cdot 3} + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.17

Multiply $5$ by $3$ .

$\frac{1}{14} + \frac{1}{15} + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.18

To write $\frac{1}{14}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\frac{1}{14} \cdot \frac{15}{15} + \frac{1}{15} + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.19

To write $\frac{1}{15}$ as a fraction with a common denominator, multiply by $\frac{14}{14}$ .

$\frac{1}{14} \cdot \frac{15}{15} + \frac{1}{15} \cdot \frac{14}{14} + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.20

Write each expression with a common denominator of $210$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.4.20.1

Multiply $\frac{1}{14}$ by $\frac{15}{15}$ .

$\frac{15}{14 \cdot 15} + \frac{1}{15} \cdot \frac{14}{14} + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.20.2

Multiply $14$ by $15$ .

$\frac{15}{210} + \frac{1}{15} \cdot \frac{14}{14} + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.20.3

Multiply $\frac{1}{15}$ by $\frac{14}{14}$ .

$\frac{15}{210} + \frac{14}{15 \cdot 14} + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.20.4

Multiply $15$ by $14$ .

$\frac{15}{210} + \frac{14}{210} + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.21

Combine the numerators over the common denominator.

$\frac{15 + 14}{210} + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.22

Add $15$ and $14$ .

$\frac{29}{210} + \frac{1}{2} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.23

One to any power is one.

$\frac{29}{210} + \frac{1}{2} (\frac{1}{2} \cdot 1 - \frac{1}{2} \cdot 0^{2})$

Step 8.4.24

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

$\frac{29}{210} + \frac{1}{2} (\frac{1}{2} - \frac{1}{2} \cdot 0^{2})$

Step 8.4.25

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

$\frac{29}{210} + \frac{1}{2} (\frac{1}{2} - \frac{1}{2} \cdot 0)$

Step 8.4.26

Multiply $0$ by $- 1$ .

$\frac{29}{210} + \frac{1}{2} (\frac{1}{2} + 0 (\frac{1}{2}))$

Step 8.4.27

Multiply $0$ by $\frac{1}{2}$ .

$\frac{29}{210} + \frac{1}{2} (\frac{1}{2} + 0)$

Step 8.4.28

Add $\frac{1}{2}$ and $0$ .

$\frac{29}{210} + \frac{1}{2} \cdot \frac{1}{2}$

Step 8.4.29

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

$\frac{29}{210} + \frac{1}{2 \cdot 2}$

Step 8.4.30

Multiply $2$ by $2$ .

$\frac{29}{210} + \frac{1}{4}$

Step 8.4.31

To write $\frac{29}{210}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{29}{210} \cdot \frac{2}{2} + \frac{1}{4}$

Step 8.4.32

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{105}{105}$ .

$\frac{29}{210} \cdot \frac{2}{2} + \frac{1}{4} \cdot \frac{105}{105}$

Step 8.4.33

Write each expression with a common denominator of $420$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.4.33.1

Multiply $\frac{29}{210}$ by $\frac{2}{2}$ .

$\frac{29 \cdot 2}{210 \cdot 2} + \frac{1}{4} \cdot \frac{105}{105}$

Step 8.4.33.2

Multiply $210$ by $2$ .

$\frac{29 \cdot 2}{420} + \frac{1}{4} \cdot \frac{105}{105}$

Step 8.4.33.3

Multiply $\frac{1}{4}$ by $\frac{105}{105}$ .

$\frac{29 \cdot 2}{420} + \frac{105}{4 \cdot 105}$

Step 8.4.33.4

Multiply $4$ by $105$ .

$\frac{29 \cdot 2}{420} + \frac{105}{420}$

Step 8.4.34

Combine the numerators over the common denominator.

$\frac{29 \cdot 2 + 105}{420}$

Step 8.4.35

Simplify the numerator.

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Step 8.4.35.1

Multiply $29$ by $2$ .

$\frac{58 + 105}{420}$

Step 8.4.35.2

Add $58$ and $105$ .

$\frac{163}{420}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{163}{420}$

Decimal Form:

$0.38809523 \dots$

Step 10