Evaluate Intergral from -inf to -4 of (1/((x+1)^2(x+2))

Answer 1

I won't provide the full answer in case this is a homework question, but think of expanding the integrand by partial fractions and then evaluate the resulting integrals with variable substitution (e.g. let u=x+1 or u=x+2). Hope this helps.

Answer 2

For the partial fraction decomposition, I get -1/(x+1) + 1/(x+1)² + 1/(x+2)

So, the given integral is found by evaluating the following limit:

lim_b→-∞ [-ln l x+1 l - 1/(x+1) + ln lx+2l]_{(b to -4)}

= lim_b→-∞ [ln l (x+2)/(x+1) l - 1 / (x+1)]_{(b to -4)}

= lim_b→-∞ [ln(2/3) + 1/3 - ( ln l (b+2)/(b+1) l - 1/ (b+1) ]

= lim_b→-∞ [ ln(2/3) + 1/3 - ln l (1+2/b) / (1 + 1/b) l - 1/(b+1) ]

= ln(2/3) + 1/3 - ln1 + 0 = ln(2/3) + 1/3

The integral converges to ln(2/3) + 1/3