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प्रश्न
Evaluate: `int_0^(1/sqrt(2)) (sin^-1x)/(1 - x^2)^(3/2) "d"x`
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उत्तर
Let I = `int_0^(1/sqrt(2)) (sin^-1x)/(1 - x^2)^(3/2) "d"x`
Put sin−1x = t
∴ x = sin t
∴ dx = cos t dt
When x = 0, t = 0 and when x = `1/sqrt(2)`, t = `pi/4`
∴ I = `int_0^(pi/4) "t"/(1 - sin^2"t")^(3/2) xx cos "t" "dt"`
= `int_0^(pi/4) "t"/(cot^2 "t")^(3/2) xx cos "t" "dt"`
= `int_0^(pi/4) "t" sec^2 "t" "dt"`
= `["t" int sec^2 "t" "dt"]_0^(pi/4) - int_0^(pi/4)["d"/("dt") ("t") int sec^2 "t" "dt"]"dt"`
= `["t"*tan "t"]_0^(pi/4) - int_0^(pi/4)1* tan "t" "dt"`
= `(pi/4* tan pi/4 - 0) - [log|sec "t"|]_0^(pi/4)`
= `pi/4(1) - [log|sec pi/4| -log|sec 0|]`
= `p/4 - (log sqrt(2) - log 1)`
= `pi/4 - (log 2^(1/2) - 0)`
∴ I = `pi/4 - 1/2 log 2`
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