Help With This Supposedly Easy Integration

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Homework Statement

Integrate: \exp(\sin(t)) / (1 + t^2)

The attempt at a solution
Ok so I tried substituting u=sin(t) du=cos(t)dt but I end up with (1 + arcsin^2(u)) on the bottom and I don't know how to integrate that.
I also tried letting t=cos(u) dt=-sin(u)du but then I end up with e^(sin(cos(t)) which I've never seen before!
If anyone knows how to do this please just give me a hint or the first step to take and I will try to do the rest! Thanks
 
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I don't think you can integrate that analytically. Is this integral part of a larger problem?
 
Ok well it was a differential equation problem that I reduced to that but here is the initial problem:

dy/dt + y\cos(t) = 1/ (1+t^2)

so I got an integrating factor of e^(sint) which led to this integral! Hope this helps maybe I did something wrong in first part.
 
Hmm, perhaps you're expected to leave the solution in terms of the integral.
 
I don't think so since the prof asked to solve it in terms of t explicitly. Maybe she made a mistake in writing the problem if this cannot be solved analytically.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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