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Question
Evaluate `int sqrt((1 + x)/(1 - x)) "d"x`, x ≠1
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Solution
Let I = `int sqrt((1 + x)/(1 - x)) "d"x`
= `int 1/sqrt(1 - x^2) "d"x + int (x"d"x)/sqrt(1 - x^2)`
= `sin^-1x + 1`
When I1 = `(x"d"x)/sqrt(1 - x^2)`.
Put 1 – x2 = t2
⇒ –2x dx = 2t dt.
Therefore I1 = – dt = – t + C
= `- sqrt(1 - x^2) + "C"`
Hence I = `sin^-1x - sqrt(1 - x^2) + "C"`.
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