Calculus Examples

$2 \int_{1}^{2} (x^{2} + 1) d x$

Step 1

Remove parentheses.

$2 \int_{1}^{2} x^{2} + 1 d x$

Step 2

Split the single integral into multiple integrals.

$2 (\int_{1}^{2} x^{2} d x + \int_{1}^{2} d x)$

Step 3

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

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

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

Simplify the answer.

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Step 6.1

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

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

Step 6.2

Substitute and simplify.

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Step 6.2.1

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

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

Step 6.2.2

Simplify.

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Step 6.2.2.1

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

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

Step 6.2.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 6.2.2.3

Combine $2$ and $\frac{3}{3}$ .

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

Step 6.2.2.4

Combine the numerators over the common denominator.

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

Step 6.2.2.5

Simplify the numerator.

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Step 6.2.2.5.1

Multiply $2$ by $3$ .

$2 (\frac{8 + 6}{3} - (\frac{1^{3}}{3} + 1))$

Step 6.2.2.5.2

Add $8$ and $6$ .

$2 (\frac{14}{3} - (\frac{1^{3}}{3} + 1))$

Step 6.2.2.6

One to any power is one.

$2 (\frac{14}{3} - (\frac{1}{3} + 1))$

Step 6.2.2.7

Write $1$ as a fraction with a common denominator.

$2 (\frac{14}{3} - (\frac{1}{3} + \frac{3}{3}))$

Step 6.2.2.8

Combine the numerators over the common denominator.

$2 (\frac{14}{3} - \frac{1 + 3}{3})$

Step 6.2.2.9

Add $1$ and $3$ .

$2 (\frac{14}{3} - \frac{4}{3})$

Step 6.2.2.10

Combine the numerators over the common denominator.

$2 \frac{14 - 4}{3}$

Step 6.2.2.11

Subtract $4$ from $14$ .

$2 (\frac{10}{3})$

Step 6.2.2.12

Combine $2$ and $\frac{10}{3}$ .

$\frac{2 \cdot 10}{3}$

Step 6.2.2.13

Multiply $2$ by $10$ .

$\frac{20}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{20}{3}$

Decimal Form:

$6.\overline{6}$

Mixed Number Form:

$6 \frac{2}{3}$

Step 8