Evaluate: ∫(2x + 1)/(x + 2)(x + 3)dx

Question

Evaluate:

\(\int\frac{(2x+1)}{(x+2)(x+3)}dx\) 

∫(2x + 1)/(x + 2)(x + 3)

Answer 1

Let I = \(\int\frac{(2x+1)}{(x+2)(x+3)}dx\),

Putting \(\frac{(2x+1)}{(x+2)(x+3)}\) = \(\frac{A}{x+2}+\frac{B}{x-3}\).......(1)

Which implies 2x=1 = A(x-3) + B(x+2) 

Now put x-3=0, x=3 

2 × 3 +1 = A(0 ) + B 3 + 2) 

So B = 7/5

Now put x + 2 = 0 ,x = -2 

- 4 +1 = A(- 2- 3) + B(0) 

So A = 3/5

From equation (1), we get,