calculate-0-2pi-1-2cost-5-4cost-dt-

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

c a l c u l a t e \int_{0}^{2 π} \frac{1 + 2 c o s t}{5 + 4 c o s t} d t

Answered by MJS last updated on 17/May/18

\int \frac{1 + 2 \cos t}{5 + 4 \cos t} d t =

[\begin{matrix} Weierstrass again : \\ u = \tan \frac{t}{2} \to d t = \frac{2 d u}{1 + u^{2}}; t = 2 \arctan u \\ \cos t = \frac{1 - u^{2}}{1 + u^{2}}; (\sin t = \frac{2 u}{1 + u^{2}}) \end{matrix}]

= - 2 \int \frac{u^{2} - 3}{(u^{2} + 1) (u^{2} + 9)} d u =

= - 2 \int (- \frac{1}{2 (u^{2} + 1)} + \frac{3}{2 (u^{2} + 9)}) d u =

= \int \frac{d u}{u^{2} + 1} - 3 \int \frac{d u}{u^{2} + 9} =

[v = \frac{u}{3} \to d u = 3 d v]

= \arctan u - 3 \int \frac{3}{9 v^{2} + 9} d v =

= \arctan u - \int \frac{d v}{v^{2} + 1} =

= \arctan u - \arctan v =

= \arctan u - \arctan \frac{u}{3} =

= \arctan (\tan \frac{t}{2}) - \arctan \frac{\tan \frac{t}{2}}{3} =

= \frac{t}{2} - \arctan (\frac{1}{3} \tan \frac{t}{2})

\underset{0}{\int^{2 π}} \frac{1 + 2 \cos t}{5 + 4 \cos t} d t = π

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