Calculus Examples

$\frac{2 u^{3}}{3} + \frac{3 u^{2}}{2}$

Step 1

Split the single integral into multiple integrals.

$\int \frac{2 u^{3}}{3} d u + \int \frac{3 u^{2}}{2} d u$

Step 2

Since $\frac{2}{3}$ is constant with respect to $u$ , move $\frac{2}{3}$ out of the integral.

$\frac{2}{3} \int u^{3} d u + \int \frac{3 u^{2}}{2} d u$

Step 3

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$\frac{2}{3} (\frac{1}{4} u^{4} + C) + \int \frac{3 u^{2}}{2} d u$

Step 4

Since $\frac{3}{2}$ is constant with respect to $u$ , move $\frac{3}{2}$ out of the integral.

$\frac{2}{3} (\frac{1}{4} u^{4} + C) + \frac{3}{2} \int u^{2} d u$

Step 5

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

$\frac{2}{3} (\frac{1}{4} u^{4} + C) + \frac{3}{2} (\frac{1}{3} u^{3} + C)$

Step 6

Simplify.

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Step 6.1

Simplify.

$\frac{2}{3} \cdot \frac{1}{4} u^{4} + \frac{3}{2} \cdot \frac{1}{3} u^{3} + C$

Step 6.2

Simplify.

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Step 6.2.1

Multiply $\frac{2}{3}$ by $\frac{1}{4}$ .

$\frac{2}{3 \cdot 4} u^{4} + \frac{3}{2} \cdot \frac{1}{3} u^{3} + C$

Step 6.2.2

Multiply $3$ by $4$ .

$\frac{2}{12} u^{4} + \frac{3}{2} \cdot \frac{1}{3} u^{3} + C$

Step 6.2.3

Cancel the common factor of $2$ and $12$ .

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Step 6.2.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{12} u^{4} + \frac{3}{2} \cdot \frac{1}{3} u^{3} + C$

Step 6.2.3.2

Cancel the common factors.

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Step 6.2.3.2.1

Factor $2$ out of $12$ .

$\frac{2 \cdot 1}{2 \cdot 6} u^{4} + \frac{3}{2} \cdot \frac{1}{3} u^{3} + C$

Step 6.2.3.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 6} u^{4} + \frac{3}{2} \cdot \frac{1}{3} u^{3} + C$

Step 6.2.3.2.3

Rewrite the expression.

$\frac{1}{6} u^{4} + \frac{3}{2} \cdot \frac{1}{3} u^{3} + C$

Step 6.2.4

Multiply $\frac{3}{2}$ by $\frac{1}{3}$ .

$\frac{1}{6} u^{4} + \frac{3}{2 \cdot 3} u^{3} + C$

Step 6.2.5

Multiply $2$ by $3$ .

$\frac{1}{6} u^{4} + \frac{3}{6} u^{3} + C$

Step 6.2.6

Cancel the common factor of $3$ and $6$ .

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Step 6.2.6.1

Factor $3$ out of $3$ .

$\frac{1}{6} u^{4} + \frac{3 (1)}{6} u^{3} + C$

Step 6.2.6.2

Cancel the common factors.

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Step 6.2.6.2.1

Factor $3$ out of $6$ .

$\frac{1}{6} u^{4} + \frac{3 \cdot 1}{3 \cdot 2} u^{3} + C$

Step 6.2.6.2.2

Cancel the common factor.

$\frac{1}{6} u^{4} + \frac{3 \cdot 1}{3 \cdot 2} u^{3} + C$

Step 6.2.6.2.3

Rewrite the expression.

$\frac{1}{6} u^{4} + \frac{1}{2} u^{3} + C$