H-0-4-tanx-dx-

Question Number 6439 by sanusihammed last updated on 27/Jun/16

H = \int_{0}^{\frac{Π}{4}} \sqrt{t a n x} d x

Commented by Temp last updated on 27/Jun/16

\int \sqrt{\tan x} d x is difficult to solve .

Commented by nburiburu last updated on 27/Jun/16

b y s u b s t i t u t i o n : t = \sqrt{t a n x} \Rightarrow t^{2} = t a n x

2 t d t = {s e c}^{2} x d x = (1 + {t a n}^{2} x) d x

d x = \frac{2 t}{1 + t^{4}} d t

\int t . \frac{2 t}{1 + t^{4}} d t = \int \frac{2 t^{2}}{1 + t^{4}} d t

a n d f r o m h e r e i t i s r a r e b u t s i m p l i e r w i t h c o m p l e x r o o t s i n a r a t i o n a l d e s c o m p o s i t i o n .

Commented by prakash jain last updated on 27/Jun/16

Question 119 has answer for \int \sqrt{\tan θ}

\frac{1}{\sqrt{2}} \tan^{- 1} (\frac{\tan θ - 1}{\sqrt{2 \tan θ}}) + \frac{1}{2 \sqrt{2}} \ln ∣ \frac{\tan θ + 1 - \sqrt{2 \tan θ}}{\tan θ + 1 + \sqrt{2 \tan θ}} ∣ + C