Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

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

Step 3

Apply the constant rule.

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

Step 4

Simplify the answer.

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Step 4.1

Combine $\frac{1}{3} x^{3}]_{0}^{2}$ and $3 x]_{0}^{2}$ .

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

Step 4.2

Substitute and simplify.

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Step 4.2.1

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

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

$\frac{1}{3} \cdot 8 + 3 \cdot 2 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0)$

Step 4.2.2.2

Combine $\frac{1}{3}$ and $8$ .

$\frac{8}{3} + 3 \cdot 2 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0)$

Step 4.2.2.3

Multiply $3$ by $2$ .

$\frac{8}{3} + 6 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0)$

Step 4.2.2.4

To write $6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{8}{3} + 6 \cdot \frac{3}{3} - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0)$

Step 4.2.2.5

Combine $6$ and $\frac{3}{3}$ .

$\frac{8}{3} + \frac{6 \cdot 3}{3} - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0)$

Step 4.2.2.6

Combine the numerators over the common denominator.

$\frac{8 + 6 \cdot 3}{3} - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0)$

Step 4.2.2.7

Simplify the numerator.

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Step 4.2.2.7.1

Multiply $6$ by $3$ .

$\frac{8 + 18}{3} - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0)$

Step 4.2.2.7.2

Add $8$ and $18$ .

$\frac{26}{3} - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0)$

Step 4.2.2.8

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

$\frac{26}{3} - (\frac{1}{3} \cdot 0 + 3 \cdot 0)$

Step 4.2.2.9

Multiply $\frac{1}{3}$ by $0$ .

$\frac{26}{3} - (0 + 3 \cdot 0)$

Step 4.2.2.10

Multiply $3$ by $0$ .

$\frac{26}{3} - (0 + 0)$

Step 4.2.2.11

Add $0$ and $0$ .

$\frac{26}{3} - 0$

Step 4.2.2.12

Multiply $- 1$ by $0$ .

$\frac{26}{3} + 0$

Step 4.2.2.13

Add $\frac{26}{3}$ and $0$ .

$\frac{26}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{26}{3}$

Decimal Form:

$8.\overline{6}$

Mixed Number Form:

$8 \frac{2}{3}$

Step 6