How to integrate e^x*cos(2x)/x^2?

To integrate e^x*cos(2x)/x^2, use integration by parts with u = cos(2x)/x^2 and dv = e^x dx.

To begin, find the derivative of u:
du/dx = (-2/x^3)cos(2x) - (2/x^2)sin(2x)

Next, find the antiderivative of dv:
v = e^x

Using the formula for integration by parts, we have:
∫ e^x*cos(2x)/x^2 dx = cos(2x)/x^2 * e^x - ∫ e^x * (-2/x^3)cos(2x) - (2/x^2)sin(2x) dx

Simplifying the integral on the right-hand side, we get:
∫ e^x*cos(2x)/x^2 dx = cos(2x)/x^2 * e^x + 2∫ e^x*sin(2x)/x^3 dx

To evaluate the remaining integral, use integration by parts again with u = sin(2x)/x^3 and dv = e^x dx. The derivative of u is:
du/dx = (6/x^4)sin(2x) - (6/x^3)cos(2x)

The antiderivative of dv is still v = e^x. Using the formula for integration by parts, we have:
∫ e^x*sin(2x)/x^3 dx = sin(2x)/x^3 * e^x - ∫ e^x * (6/x^4)sin(2x) - (6/x^3)cos(2x) dx

Simplifying the integral on the right-hand side, we get:
∫ e^x*sin(2x)/x^3 dx = sin(2x)/x^3 * e^x - 6∫ e^x*cos(2x)/x^4 dx

Substituting this result back into the original equation, we get:
∫ e^x*cos(2x)/x^2 dx = cos(2x)/x^2 * e^x + 2(sin(2x)/x^3 * e^x - 6∫ e^x*cos(2x)/x^4 dx)

Solving