How do you integrate : xdx / (x^4 + a^4)

take x^2 = t
or 2xdx = dt
or xdx=dt/2

now substitute this in ur equation :
= (xdx)/( (x^2)^2+ (a^2)^2)
= (dt/2)/( t^2 + d^2) { take a^2 =d....as because a is a constant so, dont pay too much attention to its power

now this is a general style where ∫dx/( t^2 + d^2 ) =(1/d) taninverse(t/d) +c

so ur answer is (1/2d)taninverse(t/d) +c....{' 2 ' due to ' 2 'in dt/2..compare with the general equation

= (1/(2*a^2)) taninverse(x^2/a^2) +c......hope this helps