3.5 Integration of trigonometric functions

1.

(a) $\displaystyle{-\cos{x} + \frac{\cos^{3}{x}}{3}}$(b) $\displaystyle{\frac{x}{2} - \frac{\sin{6x}}{12}}$(c) $\displaystyle{\frac{\cos^{5}{x}}{5} - \frac{\cos^{3}{x}}{3}}$

(d) $\displaystyle{\frac{1}{2}[\frac{-\cos{5x}}{5} + \cos{x}]}$(e) $\displaystyle{-\cos{x} + \frac{2\cos^{3}{x}}{3} - \frac{\cos^{5}{x}}{5}}$(f) $\displaystyle{\frac{1}{\pi}\tan{\pi x}}$

(g) $\displaystyle{\log\vert\cos{x}\vert + \frac{1}{2}\sec^{2}{x}}$(h) $\displaystyle{\frac{1}{3}\tan^{3}{x}}$(i) $\displaystyle{\frac{\sec^{5}{x}}{5} - \frac{\sec^{3}{x}}{3}}$

(j) $\displaystyle{\frac{1}{4}[\sec^{3}{x}\tan{x} + \frac{3}{2}(\sec{x}\tan{x} + \log\vert\sec{x} + \tan{x}\vert)}$(k) $\displaystyle{\frac{2\sqrt{5}}{5}\tan^{-1}(\sqrt{5}\tan{\frac{x}{2}})}$

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

(n) $\displaystyle{\frac{1}{4}\log{\vert\frac{\tan{x} - 1}{\tan{x} + 1}\vert} + \frac{x}{2}}$ (o) $\displaystyle{\frac{1}{2}[x + \log{\vert\cos{x} + \sin{x}\vert}]}$