How to integrate e^x*sin(x)/x?

To integrate e^x*sin(x)/x, use integration by parts with u = sin(x)/x and dv = e^x dx.

Integrating by parts, we have:

∫ e^x*sin(x)/x dx = ∫ u dv
= u*e^x - ∫ v du
= sin(x)/x * e^x - ∫ (cos(x)/x - sin(x)/x^2) * e^x dx

Using integration by parts again with u = cos(x)/x and dv = e^x dx, we have:

∫ e^x*sin(x)/x dx = sin(x)/x * e^x - cos(x)/x * e^x - ∫ (2/x^3 - 2/x^2 + 1/x) * e^x dx

Using integration by parts one more time with u = 2/x^3 - 2/x^2 + 1/x and dv = e^x dx, we have:

∫ e^x*sin(x)/x dx = sin(x)/x * e^x - cos(x)/x * e^x - (2/x^3 - 4/x^2 + 4/x - 4) * e^x + C

Therefore, the final answer is:

∫ e^x*sin(x)/x dx = sin(x)/x * e^x - cos(x)/x * e^x - (2/x^3 - 4/x^2 + 4/x - 4) * e^x + C