Question

integration of 3 cos x + 3 sin x upon 4 sin x + 5 cos x

Answer 1

We want Integral of 3 (Cos x + Sin x) /(4 sin x + 5 Cos x)  dx

Let   f1 = Cos x + Sin x   , and
        f2 = 4 Sin x + 5 Cos x
        f2' = d(f(x))/dx = 4 cos x - 5 sin x

Let  f1 = A f2 + B f2'
 So  Cos x + sin x = 4A sin x+ 5A cosx + 4B cosx - 5B sinx

  Compare coefficients on both sides:
            4A - 5 B = 1    and   5 A + 4 B = 1
Solving these:   A = 9/41  and   B = -1/41

So the integrand: 3 f(x) / f2(x) = 3 * 9/41 f2(x)/f2(x) - 3 *1/41 * f2'(x)/f2(x)

Solution of the Integral :  27/31 * x   - 3/41 * Ln (f2(x) )  + K
           Answer:  27x/41 - 3/41 * Ln(4 Sinx + 5 Cos x)  + K