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Questions
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Evaluate:
`inte^(sin-1) x ((x+sqrt(1-x^2))/(sqrt(1-x^2)))dx`
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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) = sin t
∴ f'(t) = cos t
∴ I = `int e^t[f(t) + f'(t)]*dt`
= et . f(t) + c
= et . sin t + c
= `e^(sin^(–1)x) * x + c`
= `x * e^(sin^(-1)x) + c`
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