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

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

To integrate e^x*cos(x)/x, we can use integration by parts. Let u = cos(x)/x and dv = e^x dx. Then du/dx = -cos(x)/x^2 - sin(x)/x^2 and v = e^x. Using the formula for integration by parts, we have:

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

Now we need to integrate ∫ e^x*cos(x)/x^2 dx and ∫ e^x*sin(x)/x^2 dx. To do this, we can use integration by parts again. Let u = 1/x^2 and dv = e^x*cos(x) dx for the first integral, and u = 1/x^2 and dv = e^x*sin(x) dx for the second integral. Then we have:

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

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

Substituting these results back into the original integral, we get:

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