Calculus Examples

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

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

Step 1

Since

3

$3$ is constant with respect to

x

$x$ , move

3

$3$ out of the integral.

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

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

Step 2

By the Power Rule, the integral of

x^{2}

$x^{2}$ with respect to

x

$x$ is

\frac{1}{3} x^{3}

$\frac{1}{3} x^{3}$ .

3 (\frac{1}{3} x^{3}]_{0}^{2})

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

Step 3

Simplify the answer.

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Step 3.1

Combine

\frac{1}{3}

$\frac{1}{3}$ and

x^{3}

$x^{3}$ .

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

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

Step 3.2

Substitute and simplify.

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Step 3.2.1

Evaluate

\frac{x^{3}}{3}

$\frac{x^{3}}{3}$ at

2

$2$ and at

0

$0$ .

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

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

Step 3.2.2

Simplify.

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Step 3.2.2.1

Raise

2

$2$ to the power of

3

$3$ .

3 (\frac{8}{3} - \frac{0^{3}}{3})

$3 (\frac{8}{3} - \frac{0^{3}}{3})$

Step 3.2.2.2

Raising

0

$0$ to any positive power yields

0

$0$ .

3 (\frac{8}{3} - \frac{0}{3})

$3 (\frac{8}{3} - \frac{0}{3})$

Step 3.2.2.3

Cancel the common factor of

0

$0$ and

3

$3$ .

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Step 3.2.2.3.1

Factor

3

$3$ out of

0

$0$ .

3 (\frac{8}{3} - \frac{3 (0)}{3})

$3 (\frac{8}{3} - \frac{3 (0)}{3})$

Step 3.2.2.3.2

Cancel the common factors.

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Step 3.2.2.3.2.1

Factor

3

$3$ out of

3

$3$ .

3 (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1})

$3 (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 3.2.2.3.2.2

Cancel the common factor.

$3 (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 3.2.2.3.2.3

Rewrite the expression.

$3 (\frac{8}{3} - \frac{0}{1})$

Step 3.2.2.3.2.4

Divide

$0$ by

$1$ .

$3 (\frac{8}{3} - 0)$

$3 (\frac{8}{3} - 0)$

$3 (\frac{8}{3} - 0)$

Step 3.2.2.4

Multiply

$- 1$ by

$0$ .

$3 (\frac{8}{3} + 0)$

Step 3.2.2.5

Add

$\frac{8}{3}$ and

$0$ .

$3 (\frac{8}{3})$

Step 3.2.2.6

Combine

$3$ and

$\frac{8}{3}$ .

$\frac{3 \cdot 8}{3}$

Step 3.2.2.7

Multiply

$3$ by

$8$ .

$\frac{24}{3}$

Step 3.2.2.8

Cancel the common factor of

$24$ and

$3$ .

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Step 3.2.2.8.1

Factor

$3$ out of

$24$ .

$\frac{3 \cdot 8}{3}$

Step 3.2.2.8.2

Cancel the common factors.

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Step 3.2.2.8.2.1

Factor

$3$ out of

$3$ .

$\frac{3 \cdot 8}{3 (1)}$

Step 3.2.2.8.2.2

Cancel the common factor.

$\frac{3 \cdot 8}{3 \cdot 1}$

Step 3.2.2.8.2.3

Rewrite the expression.

$\frac{8}{1}$

Step 3.2.2.8.2.4

Divide

$8$ by

$1$ .

$8$

$8$

$8$

$8$

$8$

$8$

Step 4

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$