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

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

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

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

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

Solution steps

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

Expand

\frac{3 x ^{2} + 2 x}{x ^{3} + x ^{2}} : \frac{3 x ^{2}}{x ^{3} + x ^{2}} + \frac{2 x}{x ^{3} + x ^{2}}

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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