Calculus Examples

$\int_{1}^{2} 2 u^{2} + 3 d u$ udu

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{2} 2 u^{2} d u + \int_{1}^{2} 3 d u$

Step 2

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$2 \int_{1}^{2} u^{2} d u + \int_{1}^{2} 3 d u$

Step 3

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

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

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

Substitute and simplify.

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Step 6.1

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

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

Step 6.2

Evaluate $3 u$ at $2$ and at $1$ .

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

Step 6.3

Simplify.

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Step 6.3.1

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

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

Step 6.3.2

One to any power is one.

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

Step 6.3.3

Combine the numerators over the common denominator.

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

Step 6.3.4

Subtract $1$ from $8$ .

$2 (\frac{7}{3}) + 3 \cdot 2 - 3 \cdot 1$

Step 6.3.5

Combine $2$ and $\frac{7}{3}$ .

$\frac{2 \cdot 7}{3} + 3 \cdot 2 - 3 \cdot 1$

Step 6.3.6

Multiply $2$ by $7$ .

$\frac{14}{3} + 3 \cdot 2 - 3 \cdot 1$

Step 6.3.7

Multiply $3$ by $2$ .

$\frac{14}{3} + 6 - 3 \cdot 1$

Step 6.3.8

Multiply $- 3$ by $1$ .

$\frac{14}{3} + 6 - 3$

Step 6.3.9

Subtract $3$ from $6$ .

$\frac{14}{3} + 3$

Step 6.3.10

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

$\frac{14}{3} + 3 \cdot \frac{3}{3}$

Step 6.3.11

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

$\frac{14}{3} + \frac{3 \cdot 3}{3}$

Step 6.3.12

Combine the numerators over the common denominator.

$\frac{14 + 3 \cdot 3}{3}$

Step 6.3.13

Simplify the numerator.

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Step 6.3.13.1

Multiply $3$ by $3$ .

$\frac{14 + 9}{3}$

Step 6.3.13.2

Add $14$ and $9$ .

$\frac{23}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{23}{3}$

Decimal Form:

$7.\overline{6}$

Mixed Number Form:

$7 \frac{2}{3}$

Step 8