∫_0^1 sin (2 tan^-1 √((1+x)/(1-x)) ) d x is equal to

Question

∫_0^1 sin (2 tan^-1 √((1+x)/(1-x)) ) d x is equal to :

MARKS Content Team · Accepted Answer

Correct answer: (B) (π)/4. Let I=∫_0^1 sin (2 tan^-1 √((1+x)/(1-x)) ) d x Put x=cos θ ⇒ d x=-sin θ d θ ∴ I=∫_π / 2^0 sin (2 tan^-1 √((1+cos θ)/(1-cos θ)) )(-sin θ) d θ =∫_0^π / 2 sin (2 tan^-1 tan ((π)/2-(θ)/2 ) ) sin θ d θ =∫_0^π / 2 sin (π-θ) sin θ d θ=∫_0^π / 2 sin^2 θ d θ =∫_0^π / 2 [(1-cos 2 θ)/2 ] d θ=1/2 [θ-(sin 2 θ)/2 ]_0^π / 2 =1/2 [(π)/2-0 ⌉=(π)/4

$\int_0^1 \sin \left(2 \tan ^{-1} \sqrt{\frac{1+x}{1-x}}\right) d x$ is equal to :

Solution

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