In class we started \int \sqrt{1+x^2} \space dx by substituting x = tan \theta and dx = sec^2 \theta \space d\theta . Note: with this substituion: \sqrt{1 + x^2} = \sqrt {1 + tan^2 \theta} = \sqrt{sec^2 \theta} = sec \space \theta .

Thus, integral becomes \int sec^3 \theta \space d\theta which we have done in class using integration by parts. Finish the integral and state in terms of x.