What is the integral of (2x^2+1)/(2x^2+x) ?

The integral of (2x^2+1)/(2x^2+x) is x-3/2 ln|2x+1|+ln|x|+1/2+C

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integral of (2x^2+1)/(2x^2+x)

\int \frac{2 x ^{2} + 1}{2 x ^{2} + x} d x

x - \frac{3}{2} ln ∣ 2 x + 1 ∣ + ln ∣ x ∣ + \frac{1}{2} + C

Solution steps

\int \frac{2 x ^{2} + 1}{2 x ^{2} + x} d x

Factor

\frac{2 x ^{2} + 1}{2 x ^{2} + x}

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

Expand

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

Apply the Sum Rule:

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

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

\int \frac{2 x ^{2}}{x ( 2 x + 1 )} d x = x + \frac{1}{2} - \frac{1}{2} ln ∣ 2 x + 1 ∣

\int \frac{1}{x ( 2 x + 1 )} d x = ln ∣ x ∣ - ln ∣ 2 x + 1 ∣

= x + \frac{1}{2} - \frac{1}{2} ln ∣ 2 x + 1 ∣ + ln ∣ x ∣ - ln ∣ 2 x + 1 ∣

Simplify

x + \frac{1}{2} - \frac{1}{2} ln ∣ 2 x + 1 ∣ + ln ∣ x ∣ - ln ∣ 2 x + 1 ∣ : x - \frac{3}{2} ln ∣ 2 x + 1 ∣ + ln ∣ x ∣ + \frac{1}{2}

= x - \frac{3}{2} ln ∣ 2 x + 1 ∣ + ln ∣ x ∣ + \frac{1}{2}

Add a constant to the solution

= x - \frac{3}{2} ln ∣ 2 x + 1 ∣ + ln ∣ x ∣ + \frac{1}{2} + C

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What is the integral of (2x^2+1)/(2x^2+x) ?
The integral of (2x^2+1)/(2x^2+x) is x-3/2 ln|2x+1|+ln|x|+1/2+C