1-pi-2-pi-3-1-sinx-cosx-dx-

Question Number 176570 by mathlove last updated on 21/Sep/22

(1) {\int_{\frac{π}{3}}}^{\frac{π}{2}} \frac{1 + s i n x}{c o s x} d x = ?

Answered by Peace last updated on 21/Sep/22

\int \frac{1 + s i n (x)}{c o s (x)} d x = \int \frac{c o s (x)}{{c o s}^{2} (x)} + \int \frac{s i n (x)}{c o s (x)} d x

= \int \frac{c o s (x)}{1 - {s i n}^{2} (x)} d x - \int \frac{d (c o s (x))}{c o s (x)}

= \int \frac{d (s i n x)}{1 - {s i n}^{2} (x)} - l n ∣ c o s (x) ∣

\underset{}{=} {a r g t h}^{-} (s i n (x)) - l n ∣ c o s (x) ∣ + c

Answered by BaliramKumar last updated on 23/Sep/22

Missing \left or extra \right

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

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

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