Branch cut contour integral - contour understanding

Question

Im trying to understand the contour when there are two poles on the real axis without poles on Im axis, because their residue cancle each other.

contour

$$I=\int_0^\infty\frac{dx}{(x+1)^2(x+2)}$$

For residue of pole $(-2)=-1$, residue of pole $(-1)=-1$.

Answer 1

When we use a contour in the complex plane to evaluate real integrals of rational functions, and the integrand has poles in the real axis, they must be first-order poles only because there is a technique to calculate the integral, but the integral doesn't exist if the order of the pole is 2 or greater.

Moreover, contour integration in the complex plane is used when it's hard to find an antiderivative of the function, or when it can't be written in terms of standard functions.

I suggest you use partial-fraction decomposition:

$$\frac{1}{(x+1)^2(x+2)} = \frac{1}{x+2} - \frac{1}{x+1} + \frac{1}{(x+1)^2}$$

and then evaluate the improper integral with the fractions on the right hand side. I hope you can work it out from here!