3.6 Integration of irrational functions

1.

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

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

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

2.

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

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

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

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

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

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

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