Calculus Examples

$\int_{0}^{3} (\frac{1}{2} x - 1) d x$

Step 1

Remove parentheses.

$\int_{0}^{3} \frac{1}{2} x - 1 d x$

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

Apply the constant rule.

$\frac{1}{2} \frac{1}{2} x^{2}]_{0}^{3} + - x]_{0}^{3}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{1}{2} x^{2}$ at $3$ and at $0$ .

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

Step 6.2

Evaluate $- x$ at $3$ and at $0$ .

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

Step 6.3

Simplify.

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Step 6.3.1

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

$\frac{1}{2} (\frac{1}{2} \cdot 9 - \frac{1}{2} \cdot 0^{2}) - 1 \cdot 3 + 0$

Step 6.3.2

Combine $\frac{1}{2}$ and $9$ .

$\frac{1}{2} (\frac{9}{2} - \frac{1}{2} \cdot 0^{2}) - 1 \cdot 3 + 0$

Step 6.3.3

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

$\frac{1}{2} (\frac{9}{2} - \frac{1}{2} \cdot 0) - 1 \cdot 3 + 0$

Step 6.3.4

Multiply $0$ by $- 1$ .

$\frac{1}{2} (\frac{9}{2} + 0 (\frac{1}{2})) - 1 \cdot 3 + 0$

Step 6.3.5

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

$\frac{1}{2} (\frac{9}{2} + 0) - 1 \cdot 3 + 0$

Step 6.3.6

Add $\frac{9}{2}$ and $0$ .

$\frac{1}{2} \cdot \frac{9}{2} - 1 \cdot 3 + 0$

Step 6.3.7

Multiply $\frac{1}{2}$ by $\frac{9}{2}$ .

$\frac{9}{2 \cdot 2} - 1 \cdot 3 + 0$

Step 6.3.8

Multiply $2$ by $2$ .

$\frac{9}{4} - 1 \cdot 3 + 0$

Step 6.3.9

Multiply $- 1$ by $3$ .

$\frac{9}{4} - 3 + 0$

Step 6.3.10

Add $- 3$ and $0$ .

$\frac{9}{4} - 3$

Step 6.3.11

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

$\frac{9}{4} - 3 \cdot \frac{4}{4}$

Step 6.3.12

Combine $- 3$ and $\frac{4}{4}$ .

$\frac{9}{4} + \frac{- 3 \cdot 4}{4}$

Step 6.3.13

Combine the numerators over the common denominator.

$\frac{9 - 3 \cdot 4}{4}$

Step 6.3.14

Simplify the numerator.

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Step 6.3.14.1

Multiply $- 3$ by $4$ .

$\frac{9 - 12}{4}$

Step 6.3.14.2

Subtract $12$ from $9$ .

$\frac{- 3}{4}$

Step 6.3.15

Move the negative in front of the fraction.

$- \frac{3}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{4}$

Decimal Form:

$- 0.75$

Step 8