For-x-gt-1-sin-1-2x-1-x-2-dx-

Question Number 37540 by rahul 19 last updated on 14/Jun/18

For x > 1,

\int \sin^{- 1} (\frac{2 x}{1 + x^{2}}) d x = ?

Answered by ajfour last updated on 14/Jun/18

l e t x = \tan θ

x > 1 \Rightarrow n π + \frac{π}{4} < θ < n π + \frac{π}{2}

\Rightarrow \frac{2 x}{1 + x^{2}} = \sin 2 θ

2 n π + \frac{π}{2} < 2 θ < 2 n π + π

\sin^{- 1} (\sin 2 θ) = \sin^{- 1} \sin (π - 2 θ)

2 m π < π - 2 θ < 2 m π + \frac{π}{2}

m = 0 i s a p p r o p r i a t e s i n c e

π - 2 θ s h o u l d l i e i n [- \frac{π}{2}, \frac{π}{2}]

\sin^{- 1} \sin (π - 2 θ) = π - 2 θ

= π - {2 \tan}^{- 1} x

I = \int (π - {2 \tan}^{- 1} x) d x

= π x - 2 (x \tan^{- 1} x - \frac{1}{2} \int \frac{2 x d x}{1 + x^{2}}) + c

I = π x - 2 x \tan^{- 1} x + \ln ∣ 1 + x^{2} ∣ + c .

Commented by rahul 19 last updated on 14/Jun/18

thank you sir. ��

Commented by rahul 19 last updated on 14/Jun/18

Ajfour sir pls try these ques \dots

Q . no . 36700 & 36096 .