sec-d-

sec-d-

Question Number 21680 by Arnab Maiti last updated on 30/Sep/17

\int \sqrt{\sec θ} d θ

Answered by alex041103 last updated on 30/Sep/17

F i r s t :

\sqrt{s e c θ} = \sqrt{\frac{1}{c o s θ}} = \frac{1}{\sqrt{c o s θ}}

W e k n o w t h a t c o s (2 θ) = 1 - 2 {s i n}^{2} (θ)

\Rightarrow \int \sqrt{s e c θ} d θ = \int \frac{d θ}{\sqrt{1 - 2 {s i n}^{2} (\frac{θ}{2})}}

W e m a k e t h e s u b s t i t u t i o n

φ = θ / 2 (2 φ = θ) a n d d θ = 2 d φ

\Rightarrow \int \sqrt{s e c θ} d θ = \int \frac{2 d φ}{\sqrt{1 - 2 {s i n}^{2} φ}}

B u t i n t e g r a l s o f t h e k i n d

\int \frac{d x}{\sqrt{1 - {k s i n}^{2} x}} a r e w e l l k n o w n t o b e

e l l i p t i c i n t e g r a l o f f i r s t k i n d

a n d \int \frac{d x}{\sqrt{1 - {k s i n}^{2} x}} = F (x ∣ k)

\Rightarrow \int \sqrt{s e c θ} d θ = 2 F (φ ∣ 2) = 2 F (θ / 2 ∣ 2) + C

A n s . \int \sqrt{s e c θ} d θ = 2 F (θ / 2 ∣ 2) + C