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

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

Integrating by parts, we have:

∫e^x*sin(x)*ln(x)dx = ln(x)*(-e^x*cos(x) - ∫-e^x*cos(x)dx)

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

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

Simplifying, we have:

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

Using integration by parts one more time with u = sin(x) and dv = e^x dx, we get:

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

Simplifying further, we have:

∫e^x*sin(x)*ln(x)dx = e^x*(-cos(x)*ln(x) + sin(x)*ln(x) - cos(x)) + C

Therefore, the integral of e^x*sin(x)*ln(x) is e^x*(-cos(x)*ln(x) + sin(x)*ln(x) - cos(x)) + C.