Calculus Examples

$\int (\frac{1}{4} v^{4} + v^{2} + v) d v$

Step 1

Remove parentheses.

$\int \frac{1}{4} v^{4} + v^{2} + v d v$

Step 2

Split the single integral into multiple integrals.

$\int \frac{1}{4} v^{4} d v + \int v^{2} d v + \int v d v$

Step 3

Since $\frac{1}{4}$ is constant with respect to $v$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int v^{4} d v + \int v^{2} d v + \int v d v$

Step 4

By the Power Rule, the integral of $v^{4}$ with respect to $v$ is $\frac{1}{5} v^{5}$ .

$\frac{1}{4} (\frac{1}{5} v^{5} + C) + \int v^{2} d v + \int v d v$

Step 5

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

$\frac{1}{4} (\frac{1}{5} v^{5} + C) + \frac{1}{3} v^{3} + C + \int v d v$

Step 6

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

$\frac{1}{4} (\frac{1}{5} v^{5} + C) + \frac{1}{3} v^{3} + C + \frac{1}{2} v^{2} + C$

Step 7

Simplify.

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Step 7.1

Simplify.

$\frac{v^{5}}{20} + \frac{v^{3}}{3} + \frac{1}{2} v^{2} + C$

Step 7.2

Reorder terms.

$\frac{1}{20} v^{5} + \frac{1}{3} v^{3} + \frac{1}{2} v^{2} + C$