0-pi-2-dx-2-cos-x-

Question Number 160597 by cortano last updated on 03/Dec/21

\int_{0}^{\frac{π}{2}} \frac{dx}{2 - \cos x} = ?

Answered by Mathspace last updated on 03/Dec/21

I = \int_{0}^{\frac{π}{2}} \frac{d x}{2 - c o s x} w e d o t h e c h a n g e m e n t

t a n (\frac{x}{2}) = t \Rightarrow I = \int_{0}^{1} \frac{2 d t}{(1 + t^{2}) (2 - \frac{1 - t^{2}}{1 + t^{2}})}

= 2 \int_{0}^{1} \frac{d t}{2 + 2 t^{2} - 1 + t^{2}} = 2 \int_{0}^{1} \frac{d t}{3 t^{2} + 1}

=_{\sqrt{3} t = u} 2 \int_{0}^{\sqrt{3}} \frac{d u}{\sqrt{3} (1 + u^{2})}

= \frac{2}{\sqrt{3}} {[a r c t a n u]}_{0}^{\sqrt{3}} = \frac{2}{\sqrt{3}} a r c t a n (\sqrt{3})

Commented by cortano last updated on 03/Dec/21

yes \dots .

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