Integration

Calculus Level 3

Evaluate

$\int_0^1 \sqrt{ \frac{1-x}{1+x} } \, dx.$

A. $\frac{\pi}{2} +1$
B. $\frac{ \pi}{2} - 1$
C. -1
D. 1

C B A D

2 solutions

Jatin Yadav
Jun 28, 2014

Hint : Put $x = \cos 2z$

rationalise the equation by multiplying with 1-x. then, it will come like integration of(1-x/rt1-x^2) substitue x as sinthita and get the answer.

Sudipan Mallick - 6 years, 11 months ago

can't we just put x=cosx, then 1-cosx=2cos^2x/2 and 1+cosx=2sin^2x/2

Saurav Zuer - 6 years, 7 months ago

Nibir Das
Jul 2, 2014

Multiply both numerator and denominator by $\sqrt { 1-x }$

and then separate as individual integrals:-

$\int { \frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } dx } \quad -\quad \int { \frac { x }{ \sqrt { 1-{ x }^{ 2 } } } dx }$

after solving the separate integrals (which is very simple) it becomes

$(\sin ^{ -1 }{ x } \quad +\quad \sqrt { 1-{ x }^{ 2 } } { ) }$

Apply the limits and the answer comes as the 2nd option.