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Question Number 35224 by abdo mathsup 649 cc last updated on 16/May/18

$$calculate{\int }_0^{2\pi }\frac{1+2cost}{5+4cost}dt$$

Answered by MJS last updated on 17/May/18

$$\int \frac{1+2\cos t}{5+4\cos t}dt=$$$$\left[\begin{array}{c}\mathrm{Weierstrass}\mathrm{again}:\\ u=\tan \frac{t}{2}\to dt=\frac{2du}{1+u^2};t=2\arctan u\\ \cos t=\frac{1-u^2}{1+u^2};(\sin t=\frac{2u}{1+u^2})\end{array}\right]$$$$=-2\int \frac{u^2-3}{(u^2+1)(u^2+9)}du=$$$$=-2\int (-\frac{1}{2(u^2+1)}+\frac{3}{2(u^2+9)})du=$$$$=\int \frac{du}{u^2+1}-3\int \frac{du}{u^2+9}=$$$$[v=\frac{u}{3}\to du=3dv]$$$$=\arctan u-3\int \frac{3}{9v^2+9}dv=$$$$=\arctan u-\int \frac{dv}{v^2+1}=$$$$=\arctan u-\arctan v=$$$$=\arctan u-\arctan \frac{u}{3}=$$$$=\arctan \left(\tan \frac{t}{2}\right)-\arctan \frac{\tan \frac{t}{2}}{3}=$$$$=\frac{t}{2}-\arctan \left(\frac{1}{3}\tan \frac{t}{2}\right)$$$$$$$$\underset{0}{\stackrel{2\pi }{\int }}\frac{1+2\cos t}{5+4\cos t}dt=\pi$$

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