Find the integral?

Question

Find the integral of 3/((x-1)(x+2)) dx from 2 to 3. 

 

Answer: ln(8/5)

 

I did partial fractions and got when x=-2, B=-1. And when x=1, A=1. What do I do next? Please show all your work.

Answer 1

3/((x-1)(x+2)) = A/(x-1) + B/(x+2) for some numbers A and B. 

Simplify both sides of the equation by multiplying by (x-1)(x+2) [we can assume x not equal to 1 and -2].

We thus get 

3 = A(x+2) + B(x-1)

Plug in x = -2 to get 3 = -3B ==> B = -1. 

Plug in x = 1 to get 3 = 3A ==> A = 1. 

 

Thus INT[3/((x-1)(x+2))]dx = INT[1/(x-1) - 1/(x+2)]dx = 

INT[1/(x-1)]dx + INT[-1/(x+2)]dx. 
Doing a U-substitution on the first one we let U = x - 1 ==> dU = dx.

Doing a V-substituion on the second one we let V = x + 2 ==> dV = dx.

 

Hence INT[1/(x-1)]dx + INT[-1/(x+2)]dx = INT[1/U]dU - INT[1/V]dV = 

ln|U| - ln|V| since INT[1/x]dx = ln|x|. 

Hence the Indefinite integral is 

ln|x-1| - ln|x+2| + C for some C in the real numbers. 

 

Now integrating from 2 to 3 should yield you ln(8/5).