Calculus Examples

$\int_{2}^{5} - 3 v + 4 d v$

Step 1

Split the single integral into multiple integrals.

$\int_{2}^{5} - 3 v d v + \int_{2}^{5} 4 d v$

Step 2

Since $- 3$ is constant with respect to $v$ , move $- 3$ out of the integral.

$- 3 \int_{2}^{5} v d v + \int_{2}^{5} 4 d v$

Step 3

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

$- 3 (\frac{1}{2} v^{2}]_{2}^{5}) + \int_{2}^{5} 4 d v$

Step 4

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

$- 3 (\frac{v^{2}}{2}]_{2}^{5}) + \int_{2}^{5} 4 d v$

Step 5

Apply the constant rule.

$- 3 (\frac{v^{2}}{2}]_{2}^{5}) + 4 v]_{2}^{5}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{v^{2}}{2}$ at $5$ and at $2$ .

$- 3 ((\frac{5^{2}}{2}) - \frac{2^{2}}{2}) + 4 v]_{2}^{5}$

Step 6.2

Evaluate $4 v$ at $5$ and at $2$ .

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

Step 6.3

Simplify.

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Step 6.3.1

Raise $5$ to the power of $2$ .

$- 3 (\frac{25}{2} - \frac{2^{2}}{2}) + 4 \cdot 5 - 4 \cdot 2$

Step 6.3.2

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

$- 3 (\frac{25}{2} - \frac{4}{2}) + 4 \cdot 5 - 4 \cdot 2$

Step 6.3.3

Cancel the common factor of $4$ and $2$ .

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Step 6.3.3.1

Factor $2$ out of $4$ .

$- 3 (\frac{25}{2} - \frac{2 \cdot 2}{2}) + 4 \cdot 5 - 4 \cdot 2$

Step 6.3.3.2

Cancel the common factors.

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Step 6.3.3.2.1

Factor $2$ out of $2$ .

$- 3 (\frac{25}{2} - \frac{2 \cdot 2}{2 (1)}) + 4 \cdot 5 - 4 \cdot 2$

Step 6.3.3.2.2

Cancel the common factor.

$- 3 (\frac{25}{2} - \frac{2 \cdot 2}{2 \cdot 1}) + 4 \cdot 5 - 4 \cdot 2$

Step 6.3.3.2.3

Rewrite the expression.

$- 3 (\frac{25}{2} - \frac{2}{1}) + 4 \cdot 5 - 4 \cdot 2$

Step 6.3.3.2.4

Divide $2$ by $1$ .

$- 3 (\frac{25}{2} - 1 \cdot 2) + 4 \cdot 5 - 4 \cdot 2$

Step 6.3.4

Multiply $- 1$ by $2$ .

$- 3 (\frac{25}{2} - 2) + 4 \cdot 5 - 4 \cdot 2$

Step 6.3.5

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

$- 3 (\frac{25}{2} - 2 \cdot \frac{2}{2}) + 4 \cdot 5 - 4 \cdot 2$

Step 6.3.6

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

$- 3 (\frac{25}{2} + \frac{- 2 \cdot 2}{2}) + 4 \cdot 5 - 4 \cdot 2$

Step 6.3.7

Combine the numerators over the common denominator.

$- 3 \frac{25 - 2 \cdot 2}{2} + 4 \cdot 5 - 4 \cdot 2$

Step 6.3.8

Simplify the numerator.

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Step 6.3.8.1

Multiply $- 2$ by $2$ .

$- 3 \frac{25 - 4}{2} + 4 \cdot 5 - 4 \cdot 2$

Step 6.3.8.2

Subtract $4$ from $25$ .

$- 3 (\frac{21}{2}) + 4 \cdot 5 - 4 \cdot 2$

Step 6.3.9

Combine $- 3$ and $\frac{21}{2}$ .

$\frac{- 3 \cdot 21}{2} + 4 \cdot 5 - 4 \cdot 2$

Step 6.3.10

Multiply $- 3$ by $21$ .

$\frac{- 63}{2} + 4 \cdot 5 - 4 \cdot 2$

Step 6.3.11

Move the negative in front of the fraction.

$- \frac{63}{2} + 4 \cdot 5 - 4 \cdot 2$

Step 6.3.12

Multiply $4$ by $5$ .

$- \frac{63}{2} + 20 - 4 \cdot 2$

Step 6.3.13

Multiply $- 4$ by $2$ .

$- \frac{63}{2} + 20 - 8$

Step 6.3.14

Subtract $8$ from $20$ .

$- \frac{63}{2} + 12$

Step 6.3.15

To write $12$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{63}{2} + 12 \cdot \frac{2}{2}$

Step 6.3.16

Combine $12$ and $\frac{2}{2}$ .

$- \frac{63}{2} + \frac{12 \cdot 2}{2}$

Step 6.3.17

Combine the numerators over the common denominator.

$\frac{- 63 + 12 \cdot 2}{2}$

Step 6.3.18

Simplify the numerator.

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Step 6.3.18.1

Multiply $12$ by $2$ .

$\frac{- 63 + 24}{2}$

Step 6.3.18.2

Add $- 63$ and $24$ .

$\frac{- 39}{2}$

Step 6.3.19

Move the negative in front of the fraction.

$- \frac{39}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{39}{2}$

Decimal Form:

$- 19.5$

Mixed Number Form:

$- 19 \frac{1}{2}$

Step 8