Calculus Examples

\int_{0}^{3} 2 x - 3 d x

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

Step 1

Split the single integral into multiple integrals.

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

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

Step 2

Since

2

$2$ is constant with respect to

x

$x$ , move

2

$2$ out of the integral.

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

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

Step 3

By the Power Rule, the integral of

x

$x$ with respect to

x

$x$ is

\frac{1}{2} x^{2}

$\frac{1}{2} x^{2}$ .

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

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

Step 4

Combine

\frac{1}{2}

$\frac{1}{2}$ and

x^{2}

$x^{2}$ .

2 (\frac{x^{2}}{2}]_{0}^{3}) + \int_{0}^{3} - 3 d x

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

Step 5

Apply the constant rule.

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

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

Step 6

Substitute and simplify.

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Step 6.1

Evaluate

\frac{x^{2}}{2}

$\frac{x^{2}}{2}$ at

3

$3$ and at

0

$0$ .

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

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

Step 6.2

Evaluate

- 3 x

$- 3 x$ at

3

$3$ and at

0

$0$ .

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

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

Step 6.3

Simplify.

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Step 6.3.1

Raise

3

$3$ to the power of

2

$2$ .

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

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

Step 6.3.2

Raising

0

$0$ to any positive power yields

0

$0$ .

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

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

Step 6.3.3

Cancel the common factor of

0

$0$ and

2

$2$ .

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Step 6.3.3.1

Factor

2

$2$ out of

0

$0$ .

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

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

Step 6.3.3.2

Cancel the common factors.

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Step 6.3.3.2.1

Factor

2

$2$ out of

2

$2$ .

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

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

Step 6.3.3.2.2

Cancel the common factor.

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

Step 6.3.3.2.3

Rewrite the expression.

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

Step 6.3.3.2.4

Divide

$0$ by

$1$ .

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

Step 6.3.4

Multiply

$- 1$ by

$0$ .

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

Step 6.3.5

Add

$\frac{9}{2}$ and

$0$ .

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

Step 6.3.6

Combine

$2$ and

$\frac{9}{2}$ .

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

Step 6.3.7

Multiply

$2$ by

$9$ .

$\frac{18}{2} - 3 \cdot 3 + 3 \cdot 0$

Step 6.3.8

Cancel the common factor of

$18$ and

$2$ .

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Step 6.3.8.1

Factor

$2$ out of

$18$ .

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

Step 6.3.8.2

Cancel the common factors.

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Step 6.3.8.2.1

Factor

$2$ out of

$2$ .

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

Step 6.3.8.2.2

Cancel the common factor.

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

Step 6.3.8.2.3

Rewrite the expression.

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

Step 6.3.8.2.4

Divide

$9$ by

$1$ .

$9 - 3 \cdot 3 + 3 \cdot 0$

Step 6.3.9

Multiply

$- 3$ by

$3$ .

$9 - 9 + 3 \cdot 0$

Step 6.3.10

Multiply

$3$ by

$0$ .

$9 - 9 + 0$

Step 6.3.11

Add

$- 9$ and

$0$ .

$9 - 9$

Step 6.3.12

Subtract

$9$ from

$9$ .

$0$

Step 7