I=∫[x+√(1+x²)]¹⁵dx/√(1+x²)

take x = tant

 on simplification we have = integral of (sect + tant )^14 sect(sect + tant) dt 

 since the derivative of sect + tant is sect(sect + tant) therefore integral of required quantity is

  (sect + tant)^15 divided by 15 now subsitute tant = x hence we have requied quantiy .