What is the integral of (3x^2-5x-17)/((x+3)(x-2)^2) ?

The integral of (3x^2-5x-17)/((x+3)(x-2)^2) is 2ln|x-2|+3/(x-2)+ln|x+3|+C

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integral of (3x^2-5x-17)/((x+3)(x-2)^2)

\int \frac{3 x ^{2} - 5 x - 17}{( x + 3 ) ( x - 2 ) ^{2}} d x

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

Solution steps

\int \frac{3 x ^{2} - 5 x - 17}{( x + 3 ) ( x - 2 ) ^{2}} d x

Apply u-substitution

= \int \frac{3 u ^{2} - 23 u + 25}{u ( u - 5 ) ^{2}} d u

Expand

\frac{3 u ^{2} - 23 u + 25}{u ( u - 5 ) ^{2}} : \frac{3 u ^{2}}{u ( u - 5 ) ^{2}} - \frac{23 u}{u ( u - 5 ) ^{2}} + \frac{25}{u ( u - 5 ) ^{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 u ^{2}}{u ( u - 5 ) ^{2}} d u - \int \frac{23 u}{u ( u - 5 ) ^{2}} d u + \int \frac{25}{u ( u - 5 ) ^{2}} d u

\int \frac{3 u ^{2}}{u ( u - 5 ) ^{2}} d u = 3 (ln ∣ u - 5 ∣ - \frac{5}{u - 5})

\int \frac{23 u}{u ( u - 5 ) ^{2}} d u = - \frac{23}{u - 5}

\int \frac{25}{u ( u - 5 ) ^{2}} d u = ln ∣ u ∣ - ln ∣ u - 5 ∣ - \frac{5}{u - 5}

= 3 (ln ∣ u - 5 ∣ - \frac{5}{u - 5}) - (- \frac{23}{u - 5}) + ln ∣ u ∣ - ln ∣ u - 5 ∣ - \frac{5}{u - 5}

Substitute back

u = x + 3

= 3 (ln ∣ x + 3 - 5 ∣ - \frac{5}{x + 3 - 5}) - (- \frac{23}{x + 3 - 5}) + ln ∣ x + 3 ∣ - ln ∣ x + 3 - 5 ∣ - \frac{5}{x + 3 - 5}

Simplify

3 (ln ∣ x + 3 - 5 ∣ - \frac{5}{x + 3 - 5}) - (- \frac{23}{x + 3 - 5}) + ln ∣ x + 3 ∣ - ln ∣ x + 3 - 5 ∣ - \frac{5}{x + 3 - 5} : 2 ln ∣ x - 2 ∣ + \frac{3}{x - 2} + ln ∣ x + 3 ∣

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

Add a constant to the solution

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

What is the integral of (3x^2-5x-17)/((x+3)(x-2)^2) ?
The integral of (3x^2-5x-17)/((x+3)(x-2)^2) is 2ln|x-2|+3/(x-2)+ln|x+3|+C