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

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

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

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

ln ∣ x ∣ - ln ∣ 2 x + 1 ∣ + C

Solution steps

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

Take the partial fraction of

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

= \int \frac{1}{x} - \frac{2}{2 x + 1} 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

= \int \frac{1}{x} d x - \int \frac{2}{2 x + 1} d x

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ - ln ∣ 2 x + 1 ∣

Add a constant to the solution

= ln ∣ x ∣ - ln ∣ 2 x + 1 ∣ + C

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

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