calculate-pi-4-pi-3-dx-cosx-sinx-

Question Number 46841 by maxmathsup by imad last updated on 01/Nov/18

c a l c u l a t e \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{d x}{c o s x s i n x}

Commented by maxmathsup by imad last updated on 01/Nov/18

w e h a v e I = \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{d x}{c o s x s i n x} = 2 \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{d x}{s i n (2 x)} =_{2 x = t} 2 \int_{\frac{π}{2}}^{\frac{2 π}{3}} \frac{1}{s i n (t)} \frac{d t}{2}

= \int_{\frac{π}{2}}^{\frac{2 π}{3}} \frac{d t}{s i n t} =_{t a n (\frac{t}{2}) = u} \int_{t a n (\frac{π}{4})}^{t a n (\frac{π}{3})} \frac{1}{\frac{2 u}{1 + u^{2}}} \frac{2 d u}{1 + u^{2}} = \int_{1}^{\sqrt{3}} \frac{d u}{u} = {[l n ∣ u ∣]}_{1}^{\sqrt{3}}

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

Answered by tanmay.chaudhury50@gmail.com last updated on 01/Nov/18

\int_{\frac{π}{4}}^{\frac{π}{3}} \frac{d x}{{c o s}^{2} x \times t a n x}

\int_{\frac{π}{4}}^{\frac{π}{3}} \frac{d (t a n x)}{t a n x}

∣ l n (t a n x) ∣_{\frac{π}{4}}^{\frac{π}{3}}

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

= \frac{1}{2} l n 3