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

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

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

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

5 ln ∣ x - 2 ∣ + 3 arctan (x) + C

Solution steps

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

Take the partial fraction of

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

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

\int \frac{5}{x - 2} d x = 5 ln ∣ x - 2 ∣

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

= 5 ln ∣ x - 2 ∣ + 3 arctan (x)

Add a constant to the solution

= 5 ln ∣ x - 2 ∣ + 3 arctan (x) + C

x \to 0 lim (\frac{x ^{2} - 3 x}{2 x})

\int_{- \infty}^{0} 15 x e^{3 x} d x

f (x) = 3 x + 4

\frac{1 + x f ( x )}{x}

\int \frac{1}{2 - 3 u - u ^{2}} d u

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