pi-6-pi-3-1-sin-2x-dx-

Question Number 64332 by Chi Mes Try last updated on 16/Jul/19

\underset{π / 6}{\int^{π / 3}} \frac{1}{\sin 2 x} d x =

Commented by mathmax by abdo last updated on 16/Jul/19

l e t A = \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{d x}{s i n (2 x)} c h a n g e m e n t 2 x = t g i v e

A = \int_{\frac{π}{3}}^{\frac{2 π}{3}} \frac{d t}{2 s i n t} =_{t a n (\frac{t}{2}) = u} \frac{1}{2} \int_{\frac{1}{\sqrt{3}}}^{\sqrt{3}} \frac{2 d u}{(1 + u^{2}) \frac{2 u}{1 + u^{2}}} = \frac{1}{2} \int_{\frac{1}{\sqrt{3}}}^{\sqrt{3}} \frac{d u}{u}

= \frac{1}{2} {[l n ∣ u ∣]}_{\frac{1}{\sqrt{3}}}^{\sqrt{3}} = \frac{1}{2} {l n (\sqrt{3}) + l n (\sqrt{3})} = \frac{l n (3)}{2}

A = \frac{l n (3)}{2} .

Commented by Tony Lin last updated on 17/Jul/19

\int_{\frac{π}{6}}^{\frac{π}{3}} \frac{1}{s i n 2 x} d x

= \int_{\frac{π}{6}}^{\frac{π}{3}} c s c 2 x d x

= - \frac{1}{2} l n ∣ c s c 2 x + c o t 2 x ∣ ∣_{\frac{π}{6}}^{\frac{π}{3}}

= - \frac{1}{2} l n ∣ c o t x ∣_{\frac{π}{6}}^{\frac{π}{3}}

= - \frac{l n (\frac{1}{\sqrt{3}})}{2} + \frac{l n (\sqrt{3})}{2}

= \frac{l n 3}{2}