U may try this.

question says Evaluate: \int_{0}^{\infty}{\frac{e^{-ax}\sin{x}}{x}}dx \\Evaluate: \int_{0}^{\infty}{\frac{\sin{x}}{x}dx}

for second we may use this \int_{a}^{\infty} f(x)dx = \lim_{b \to \infty} \int_{a}^{b} f(x)dx property

but it cannot be evaluated directly.
do try the first one.

First one:
Let I = the integral
calculate dI/da ...

it will be simplified to deal with it..

after u get the result
put a=0 in the result to get answer of second part

If you cant solve.. ill copy paste the method later

well done, asish.

found the values?