Question

Please give integration of this urgently

Answer 1

let x^2 = y
dy =2xdx
now
inte{√coty}dy
now,let coty= z^2
-cosec^2ydy = 2zdz
dy = -2zdz/1+z^4
now,put value of coty & dy
inte{2z^2/1+z^4}dz
inte{[(z^2+1) + (z^2-1)]/(1+z^4)}dz [dividing numenetor &denominator by z^2]
inte{(1+1/z^2)/(z^2+1/z^2)}dz + inte{(1-1/z^2)/(z^2+1/z^2)dz .......(1)
now let (z-1/z)=k , so d(k) = (1+1/z^2)dz
&(z^2+(1/z^2)) = k^2 -2
now let (z+1/z)= t,so d(t) = (1-1/z^2)dz &
(z^2+(1/z^2)) = t^2 +2
NOW PUT THESE VALUES IN INITE (1)
inte{dk/(k^2-2)} +inte{dt/(t^2 +2)}
1/2√2ln(k-√2/k+√2) +1/√2tan(inverse)(t/√2)+C

now put value of k&t as k= (z-1/z) & z = √cotx^2
and t = z+1/z