Solving an Integral Without Software: Step-by-Step

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I want you to solve me a following problem without any software and explain me step to step:

int(exp(-x)*sin(a*x)/(x),x=0..infinity);
thenk you
 
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Thread moved to the homework forums. Welcome to the PF, mashjavad. Please tell us what you know already about integration. What learning resources do you have available?
 
Hi mashjavad,

Looks like integration by parts doesn't solve the integral. It takes the form (e^-x sin ax)/x. I used mathematical software to evaluate and obtained exponential integrals with complex numbers in the result. I tink this involves incomplete gamma functions or numerical methods of integration.

Sorry couldn't help much.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

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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