What is the integral of (x^3+2x^5)/(2x) ?

The integral of (x^3+2x^5)/(2x) is (x^3)/6+(x^5)/5+C

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integral of (x^3+2x^5)/(2x)

\int \frac{x ^{3} + 2 x ^{5}}{2 x} d x

\frac{x ^{3}}{6} + \frac{x ^{5}}{5} + C

Solution steps

\int \frac{x ^{3} + 2 x ^{5}}{2 x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{x ^{3} + 2 x ^{5}}{x} d x

Expand

\frac{x ^{3} + 2 x ^{5}}{x} : \frac{x ^{3}}{x} + \frac{2 x ^{5}}{x}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int \frac{x ^{3}}{x} d x + \int \frac{2 x ^{5}}{x} d x)

\int \frac{x ^{3}}{x} d x = \frac{x ^{3}}{3}

\int \frac{2 x ^{5}}{x} d x = \frac{2 x ^{5}}{5}

= \frac{1}{2} (\frac{x ^{3}}{3} + \frac{2 x ^{5}}{5})

Simplify

\frac{1}{2} (\frac{x ^{3}}{3} + \frac{2 x ^{5}}{5}) : \frac{x ^{3}}{6} + \frac{x ^{5}}{5}

= \frac{x ^{3}}{6} + \frac{x ^{5}}{5}

Add a constant to the solution

= \frac{x ^{3}}{6} + \frac{x ^{5}}{5} + C

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What is the integral of (x^3+2x^5)/(2x) ?
The integral of (x^3+2x^5)/(2x) is (x^3)/6+(x^5)/5+C