How to integrate $\frac{1}{x\sqrt{1+x^2}}$ using substitution? How you integrate 
$$\frac{1}{x\sqrt{1+x^2}}$$
using following substitution? $u=\sqrt{1+x^2} \implies du=\dfrac{x}{\sqrt{1+x^2}}\, dx$ 
And now I don't know how to proceed using substitution rule.

Let $x=\tan\theta$, then $dx=\sec^2\theta\,d\theta$, now
$$\int{dx\over x\sqrt{1+x^2}}=\int{\sec\theta\,d\theta\over\tan\theta}=\int\csc\theta\,d\theta.$$
Can you take it from here?
