Integrate √(4+x^2)/x dx

∫√(4 + x²)/x dx

Let x = 2*tan(t)

∫√(4 + 4*tan²(t))*2sec²(t) dt/(2*tan(t))

2*∫sec^3(t)/tan(t) dt

2*∫(sec(t) + sec(t)*tan²(t))/tan(t) dt

2*∫csc(t) dt + 2sec(t) + C

-2*ln|csc(t) + cot(t)| + 2*sec(t) + C

Answer: √(4 + x²) - 2*ln|(2 + √(4 + x²))/x| + C

x^2/2 + 4 Log[x] + c

4log x =x^2/2+c