What is the integral of (4x^3+5x)/(2x) ?

The integral of (4x^3+5x)/(2x) is 1/2 ((4x^3)/3+5x)+C

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integral of (4x^3+5x)/(2x)

\int \frac{4 x ^{3} + 5 x}{2 x} d x

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

Solution steps

\int \frac{4 x ^{3} + 5 x}{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{4 x ^{3} + 5 x}{x} d x

Expand

\frac{4 x ^{3} + 5 x}{x} : \frac{4 x ^{3}}{x} + \frac{5 x}{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{4 x ^{3}}{x} d x + \int \frac{5 x}{x} d x)

\int \frac{4 x ^{3}}{x} d x = \frac{4 x ^{3}}{3}

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

= \frac{1}{2} (\frac{4 x ^{3}}{3} + 5 x)

Add a constant to the solution

= \frac{1}{2} (\frac{4 x ^{3}}{3} + 5 x) + C

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What is the integral of (4x^3+5x)/(2x) ?
The integral of (4x^3+5x)/(2x) is 1/2 ((4x^3)/3+5x)+C