.

3.6 Integration of irrational functions

1.

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

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

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

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

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