Evaluate the indefinite integral: ∫ (e^x)sin(x) dx

Note: for what’s about to come, ' denotes derivative with respect to x. Method 1 (integration by parts): let I = (e^x)sin(x) dx let u = sin(x), u' = cos(x) and v' = e^x = v then integrate by parts, => I = uv - ∫u’v dx, substitute in u,v and u’ obtains: I =(e^x)sin(x) - ∫ (e^x)cos(x) dx () Now integrate by parts ∫ (e^x)cos(x) dx: let a = cos(x), a' = -sin(x) and b' = b = e^x hence: ∫ (e^x)cos(x) dx =(e^x)cos(x) + I, substitute into (): I = (e^x)(sin(x) + cos(x)) - I + C, for some constant C => I = 1/2(e^x)(sin(x) + cos(x)) + C  Method 2 (compare coefficients): Let y = (e^x)(Acos(x) + Bsin(x)), for some constants A, B to be determined => y' =(e^x)((A+B)cos(x) + (B-A)sin(x)) ≡(e^x)sin(x), hence by comparing coefficients, we have: B - A = 1 and A + B = 0 => B = 1/2, A = -1/2 => y = (1/2)(e^x)(sin(x) + cos(x)) + D, for some constants D