Solve-integral-of-sec-x-dx-given-that-t-tan-x-2-

Question Number 5866 by sanusihammed last updated on 02/Jun/16

S o l v e i n t e g r a l o f s e c (x) d x . g i v e n t h a t t = t a n (\frac{x}{2})

Answered by Yozzii last updated on 02/Jun/16

L e t t = t a n 0.5 x \Rightarrow d t = 0.5 {s e c}^{2} 0.5 x d x

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

c o s x = \frac{1 - t^{2}}{1 + t^{2}} \Rightarrow s e c x = \frac{1 + t^{2}}{1 - t^{2}}

∴ \int s e c x d x = \int \frac{1 + t^{2}}{1 - t^{2}} \times \frac{2}{1 + t^{2}} d t

= \int (\frac{1}{1 - t} + \frac{1}{1 + t}) d t

= l n ∣ 1 + t ∣ - l n ∣ 1 - t ∣ + C

= l n ∣ \frac{1 + t}{1 - t} ∣ + C

\int s e c x d x = l n ∣ \frac{1 + t a n 0.5 x}{1 - t a n 0.5 x} ∣ + C

Commented by sanusihammed last updated on 02/Jun/16

I n t e r e s t i n g t h a n k s .