3.6 Integration of irrational functions

1.

(a) $\displaystyle{2\sqrt{x} - 2\log{\vert 1+\sqrt{x}\vert} + C}$ (b) $\displaystyle{x - 2\sqrt{x} + 2\log\vert 1 + \sqrt{x}\vert+ C}$

(c) $\displaystyle{\frac{1}{2}\log{\vert\frac{\sqrt{1 - e^{x}} - 1}{\sqrt{1 - e^{x}} + 1}\vert} + C}$

(d) $\displaystyle{\frac{2}{3}(x-1)^{\frac{3}{2}} + 2\sqrt{x-1} + C}$ (e) $\displaystyle{\sqrt{x^{2} + 4} + C}$

(f) $\displaystyle{\frac{(x^{2} + 4)^{3/2}}{3} - 4\sqrt{x^{2} + 4} + C}$ (g) $\displaystyle{\log{\vert x - 1 + \sqrt{x^{2} - 2x -3}\vert} + C}$

(h) $\displaystyle{2(\sqrt{x^{2} - 1} - \tan^{-1}(\sqrt{x^{2} - 1})) + C}$