$$\int\limits_0^\infty \frac{x^2dx}{(x^4+4)^2} $$
In my complex analysis class problem list I have this integral, it's supposed to be solved by residue theorem. My first attempt was to apply it directly, but calculations in poles were too bad. Then I tried to use substitute, but failed. Looks like it's impossible to make this function a sum of elementary fractions.
So currently I'm looking for a proper substitute or a trick which would let me to use residue theorem without troubles,thanks.
Best Answer
Consider the similar integral
$$I=\int_0^\infty \frac{x^2}{(x^4+1)^2}dx$$
Under the variable interchange $x\leftrightarrow\frac{1}{x}$ we get
$$I = \int_0^\infty\frac{x^4}{(x^4+1)^2}dx = \int_0^\infty \frac{1}{x^4+1}-\frac{1}{(x^4+1)^2}dx$$
It is easier to apply residue theorem to the simple pole and the order two pole with a trivial numerator. Can you think of how to apply this to your integral?