find-pi-6-pi-3-dx-cos-x-sin-x-

Question Number 34282 by math khazana by abdo last updated on 03/May/18

f i n d \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{d x}{c o s (x) s i n (x)}

Commented by math khazana by abdo last updated on 07/May/18

I = \int_{\frac{π}{6}}^{\frac{π}{3}} 2 \frac{d x}{s i n (2 x)} = 2 \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{d x}{s i n (2 x)}

= 2 {[\frac{1}{2} l n ∣ t a n x ∣]}_{\frac{π}{6}}^{\frac{π}{3}} = l n (t a n (\frac{π}{3})) - l n (t a n (\frac{π}{6}))

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