Integrate the following functions w.r.t. x : esin-1x.[x+1-x21-x2] - Mathematics and Statistics

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Sum

Integrate the following functions w.r.t. x : `e^(sin^-1x).[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`

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Solution

Let I = `int e^(sin^-1x)[(x + sqrt(1 - x^2))/sqrt(1 - x^2)].dx`

= `int e^(sin^-1x) [x + sqrt(1 - x^2)].(1)/sqrt(1 - x^2).dx`

Put sin–1 x = t

∴ `(1)/sqrt(1 - x^2).dx` = dt

and x = sin t

∴ I = `int e^t [sin t + sqrt(1 - sin^2 t].dt`

= `int e^t [sin t + sqrt(cos^2t].dt`

= `int e^t(sin t + cos t).dt`

Let f(t) = sint
∴ f'(t) = cost

∴ I = `int e^t[f(t) + f'(t)].dt`

= et . f(t) + c
= et . sin t + c
= `e^(sin^–1_x) . x + c`.

  Is there an error in this question or solution?
Chapter 3: Indefinite Integration - Exercise 3.3 [Page 138]

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