Question

What is integral of (ax+b)/(cx+d)

Answer 1

Let t = cx + d
x = (t-d)/c
dx = dt/c
Then
ax + b = a[(t-d)/c] + b = at/c -ad/c + b
= at/c + b - ad/c
So, (ax+b)/(cx+d)dx = [(at/c + b - ad/c)dt]/(tc) = [(a/c^2 + (bc - ad)/(tc^2)]dt Integrate {[(a/c^2 + (bc - ad)/(tc^2)]dt}
= (a/c^2)t + [(bc - ad)/(c^2)] ln(t)
Replace t with cx + d to get the answer!