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

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

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integral of (2x^2+x-1)/(2x^3+x^2-5x+2)

\int \frac{2 x ^{2} + x - 1}{2 x ^{3} + x ^{2} - 5 x + 2} d x

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

Solution steps

\int \frac{2 x ^{2} + x - 1}{2 x ^{3} + x ^{2} - 5 x + 2} d x

Take the partial fraction of

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

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

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

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

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

Add a constant to the solution

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

\int \frac{x - 1}{x ^{2} - 4} d x

\int \frac{9}{x ^{2} - 16} d x

x \to 2 - lim (x^{3})

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

y^{^{''}} + 36 y = 0, y (0) = 1, y^{^{'}} (0) = 0

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