What is the integral of (3x+4)/((x+1)(x+2)) ?

The integral of (3x+4)/((x+1)(x+2)) is 2ln|x+2|+ln|x+1|+C

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integral of (3x+4)/((x+1)(x+2))

\int \frac{3 x + 4}{( x + 1 ) ( x + 2 )} d x

2 ln ∣ x + 2 ∣ + ln ∣ x + 1 ∣ + C

Solution steps

\int \frac{3 x + 4}{( x + 1 ) ( x + 2 )} d x

Apply u-substitution

: \int \frac{3 u + 1}{u ( u + 1 )} d u

= \int \frac{3 u + 1}{u ( u + 1 )} d u

Expand

\frac{3 u + 1}{u ( u + 1 )} : \frac{3 u}{u ( u + 1 )} + \frac{1}{u ( u + 1 )}

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}{u ( u + 1 )} d u + \int \frac{1}{u ( u + 1 )} d u

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

\int \frac{1}{u ( u + 1 )} d u = ln ∣ u ∣ - ln ∣ u + 1 ∣

= 3 ln ∣ u + 1 ∣ + ln ∣ u ∣ - ln ∣ u + 1 ∣

Substitute back

u = x + 1

= 3 ln ∣ x + 1 + 1 ∣ + ln ∣ x + 1 ∣ - ln ∣ x + 1 + 1 ∣

Simplify

= 2 ln ∣ x + 2 ∣ + ln ∣ x + 1 ∣

Add a constant to the solution

= 2 ln ∣ x + 2 ∣ + ln ∣ x + 1 ∣ + C

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What is the integral of (3x+4)/((x+1)(x+2)) ?
The integral of (3x+4)/((x+1)(x+2)) is 2ln|x+2|+ln|x+1|+C