3.6 Integration of irrational functions

1.

(a) $${ 2[\frac{(1 + x)^{5/2}}{5} - \frac{(1+x)^{3/2}}{3}]}$$(b) $${x + 2\sqrt{x} + 2\log{\vert\sqrt{x} - 1\vert}}$$

(c) $${\log{\vert\frac{\sqrt{1+e^x} -1}{\sqrt{1+e^{x} + 1}}\vert}}$$(d) $${\frac{2(x-1)^{7/2}}{7} + \frac{4(x-1)^{5/2}}{5} + \frac{2(x-1)^{3/2}}{3}}$$

(e) $${\frac{1}{2}(x-1)^2 \sqrt{\frac{x+1}{x-1}} + \log\left(1 + \sqrt{\frac{x+1}{x-1}}\right) - \log\left(1 - \sqrt{\frac{x+1}{x-1}}\right)}$$

2.

(a) $${{\sqrt{-4 + {x^2}}}}$$(b) $${2[\sqrt{4 - x^2} - \frac{x}{2}\frac{\sqrt{4 - x^2}}{2}}$$

(c) $${\frac{1}{6}\log\vert\frac{3 + e^{x}}{3 - e^{x}}\vert}$$(d) $${-\frac{(1 - x^2)^{3/2}}{3x^3}}$$

(e) $${\frac{\sqrt{x^2 - a^2}}{a^{2}x}}$$(f) $${-\frac{\sqrt{4 + e^{2x}}}{4e^{x}}}$$(g) $${\frac{1}{2}\log\vert\frac{1+\cos{t}}{1 - \cos{t}}\vert}$$

(h) $${-3\sin^{-1}\left(\frac{3-x}{3}\right)- \sqrt{6x - x^2}}$$

(i) $${\sqrt{x^2 - 2x - 3} + \log\vert\frac{x-1 + \sqrt{(x-1)^2 - 4}}{2}\vert}$$

(j) $${-\frac{\sin^{-1}{(3-x)}}{2} - \frac{(3-x)\sqrt{1 - (3-x)^2}}{2}}$$

(k) $${{\sqrt{x \left( 6 + x \right) }} \left( -\frac{9}{2} + \frac{... ...}{3} \right) + \frac{27}{2} \log (3 + x + {\sqrt{x \left( 6 + x \right)}})}$$