I'm working on tough integrals that basically contain a fraction inside them. Here's a (simplified) example:
$\int_{-\pi}^\pi{\frac{1+e^{i t}}{e^{i t}}dt}$
I'm interested in solving this using differentiation under the integral, and I'm hoping that someone can help me. My work so far follows...
First, I restate the integral as follows:
$-i\int_{-\pi}^\pi{\frac{1+e^{i t}}{e^{i t}}\frac{ie^{i t}}{e^{i t}}dt}= -i\int_{|z|=1}{\frac{1+z}{z}\frac{1}{z}dz}$
Here's where we can use differentiation under the integral, even if it seems too much for such a simple problem.
The integral is now: $-i\int_{|z|=1}{\frac{1+z}{z^2}dz}$
and the fractions in some of the problems that I'm working on will be extremely hard to work with. So I set up a new function:
$F(y)=-i\int_{|z|=1}{(1+z)\sin{\left(\frac{y}{z^2}-\frac{1}{z^2}\right)}dz}$
Note that taking the derivative w.r.t. $y$ yields the result I'm looking for, if $y$ is then set equal to $1$:
F'(y)=-i\int_{|z|=1}{\frac{1+z}{z^2}\cos{\left(\frac{y}{z^2}-\frac{1}{z^2}\right)}dz}
F'(1)=-i\int_{|z|=1}{\frac{1+z}{z^2}\cos{0}dz}=F'(1)=-i\int_{|z|=1}{\frac{1+z}{z^2}dz}
This method then seems extremely easy to do, even with far more complicated divisors in the fractions, since I can use Cauchy to integrate, and then proceed with differentiation under the integral. I'd like to know if this seems correct so far. I'm wondering if there are additional considerations I need to be aware of before getting too complex. But it seems that I have a chance of "building" integrals in this way, so I want to be sure that I'm correct.
If this seems correct and checks out, I'd really like to know, and maybe this is too much for this question, if I can use this with multiple fractions and multiple variables. For instance, would a function like $\int{\frac{p(x)}{q(x)r(x)}}$ be easy to adapt to this method?