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

The integral of (2x^4+3x)/(2x^2) is (x^3)/3+3/2 ln|x|+C

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

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

\frac{x ^{3}}{3} + \frac{3}{2} ln ∣ x ∣ + C

Solution steps

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

Simplify

\frac{2 x ^{4} + 3 x}{x ^{2}} : \frac{2 x ^{3} + 3}{x}

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

Expand

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

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

\int 2 x^{2} d x = \frac{2 x ^{3}}{3}

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

Simplify

\frac{1}{2} (\frac{2 x ^{3}}{3} + 3 ln ∣ x ∣) : \frac{x ^{3}}{3} + \frac{3}{2} ln ∣ x ∣

= \frac{x ^{3}}{3} + \frac{3}{2} ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{3}}{3} + \frac{3}{2} ln ∣ x ∣ + C

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