What is the integral of 1/(2x^2+5x+2) ?

The integral of 1/(2x^2+5x+2) is 1/3 ln|2x+1|-1/3 ln|x+2|+C

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integral of 1/(2x^2+5x+2)

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

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

Solution steps

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

Take the partial fraction of

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

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

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

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

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

Add a constant to the solution

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

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