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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

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

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

Simplify

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

Add a constant to the solution

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

x \to 0 lim (\frac{4 x}{x})

3 t \cdot cos (2 t)

\int 3 sin^{3} (6 x) cos^{3} (6 x) d x

\frac{d}{d t} (d cos (t) + t^{2} sin (t))

\frac{d}{d x} (r (x) e^{r (x) x})

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