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Question
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
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Solution
Let I = `int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
Put sin−1 x = t .......(i)
∴ x = sin t
Differentiating (i) w.r.t. x, we get
`1/sqrt(1 -x^2) "d"x` = dt
∴ I = `int "e"^"t"[sin "t" + sqrt(1 - sin^2"t")] "dt"`
= `int "e"^"t" [sin "t" + cos "t"] "dt"`
Put f(t) = sin t
∴ f'(t) = cos t
∴ I = `int"e"^"t"["f"("t") + "f'"("t")] "dt"`
= et f(t) + c
= et sin t + c
∴ I = `"e"^(sin^(-1_x)) (x) + "c"`
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