Calculus Examples

$\int_{0}^{5} [(x + 2) - \frac{x}{2}] d x$

Step 1

Remove parentheses.

$\int_{0}^{5} (x + 2) - \frac{x}{2} d x$

Step 2

Simplify.

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Step 2.1

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

$\int_{0}^{5} x \cdot \frac{2}{2} - \frac{x}{2} + 2 d x$

Step 2.2

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

$\int_{0}^{5} \frac{x \cdot 2}{2} - \frac{x}{2} + 2 d x$

Step 2.3

Combine the numerators over the common denominator.

$\int_{0}^{5} \frac{x \cdot 2 - x}{2} + 2 d x$

Step 2.4

Move $2$ to the left of $x$ .

$\int_{0}^{5} \frac{2 \cdot x - x}{2} + 2 d x$

Step 2.5

Subtract $x$ from $2 x$ .

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

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

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

Step 6

Apply the constant rule.

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

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{1}{2} x^{2}$ at $5$ and at $0$ .

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

Step 7.2

Evaluate $2 x$ at $5$ and at $0$ .

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

Step 7.3

Simplify.

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Step 7.3.1

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

$\frac{1}{2} (\frac{1}{2} \cdot 25 - \frac{1}{2} \cdot 0^{2}) + 2 \cdot 5 - 2 \cdot 0$

Step 7.3.2

Combine $\frac{1}{2}$ and $25$ .

$\frac{1}{2} (\frac{25}{2} - \frac{1}{2} \cdot 0^{2}) + 2 \cdot 5 - 2 \cdot 0$

Step 7.3.3

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

$\frac{1}{2} (\frac{25}{2} - \frac{1}{2} \cdot 0) + 2 \cdot 5 - 2 \cdot 0$

Step 7.3.4

Multiply $0$ by $- 1$ .

$\frac{1}{2} (\frac{25}{2} + 0 (\frac{1}{2})) + 2 \cdot 5 - 2 \cdot 0$

Step 7.3.5

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

$\frac{1}{2} (\frac{25}{2} + 0) + 2 \cdot 5 - 2 \cdot 0$

Step 7.3.6

Add $\frac{25}{2}$ and $0$ .

$\frac{1}{2} \cdot \frac{25}{2} + 2 \cdot 5 - 2 \cdot 0$

Step 7.3.7

Multiply $\frac{1}{2}$ by $\frac{25}{2}$ .

$\frac{25}{2 \cdot 2} + 2 \cdot 5 - 2 \cdot 0$

Step 7.3.8

Multiply $2$ by $2$ .

$\frac{25}{4} + 2 \cdot 5 - 2 \cdot 0$

Step 7.3.9

Multiply $2$ by $5$ .

$\frac{25}{4} + 10 - 2 \cdot 0$

Step 7.3.10

Multiply $- 2$ by $0$ .

$\frac{25}{4} + 10 + 0$

Step 7.3.11

Add $10$ and $0$ .

$\frac{25}{4} + 10$

Step 7.3.12

To write $10$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{25}{4} + 10 \cdot \frac{4}{4}$

Step 7.3.13

Combine $10$ and $\frac{4}{4}$ .

$\frac{25}{4} + \frac{10 \cdot 4}{4}$

Step 7.3.14

Combine the numerators over the common denominator.

$\frac{25 + 10 \cdot 4}{4}$

Step 7.3.15

Simplify the numerator.

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Step 7.3.15.1

Multiply $10$ by $4$ .

$\frac{25 + 40}{4}$

Step 7.3.15.2

Add $25$ and $40$ .

$\frac{65}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{65}{4}$

Decimal Form:

$16.25$

Mixed Number Form:

$16 \frac{1}{4}$

Step 9