how to integrate e^xsinxcosxdx

Question

Question Evaluate it by parts add a word representing all the functions are defined

Answer 1

Let I=∫eˣsin(x)cos(x)dx=½∫eˣsin(2x)dx.

Integrate by parts: Let u=eˣ, du=eˣdx, dv=sin(2x)dx, v=-cos(2x)/2.

I=½(-½eˣcos(2x)+½∫eˣcos(2x)dx)

Integrate by parts again: Let J=½∫eˣcos(2x)dx, dv=cos(2x)dx, v=½sin(2x) and

J=½(½eˣsin(2x)-½∫eˣsin(2x)dx)=¼eˣsin(2x)-½I and

I=½(-½eˣcos(2x)+½∫eˣcos(2x)dx)=½(-½eˣcos(2x)+J)

So I=½(-½eˣcos(2x)+¼eˣsin(2x)-½I)=-¼eˣcos(2x)+⅛eˣsin(2x)-¼I

5I/4=-¼eˣcos(2x)+⅛eˣsin(2x), 5I=-eˣcos(2x)+½eˣsin(2x)

I=⅕(½eˣsin(2x)-eˣcos(2x))+C

I=⅕eˣ(sin(x)cos(x)-2cos²(x)+1)+C or ⅕eˣ(sin(x)cos(x)+2sin²(x)-1)+C, where C is a constant.

CHECK

dI/dx=⅕(eˣcos(2x)+½eˣsin(2x)+2eˣsin(2x)-eˣcos(2x))=½eˣsin(2x)=eˣsin(x)cos(x).