How generalize the following?
Evaluating two improper integrals involving powers on $\sin(t)$
How to generalize the above for higher powers of sine(different powers for the sine function, since raising to the same power yields a zero) and higher powers of the variable $t$ in the denominator?
$$\int_0^{\infty} \frac{\sin^a (t)\left(\sin ^b(t)-t^c\right)}{t^d} d t.$$
For the integral to converge, $d<a, b, c.$
I tried using the Laplace transform approach based on the third answer and used the formula for the powers of the sine function, but when I reach the stage to apply polynomial long division it gets too complicated.
Any idea how to approach the above?
Best Answer
Usually these kinds of integrals are solved by applying complex analysis using the residue theorem: https://ekamperi.github.io/math/2020/12/15/cauchy-residue-theorem.html