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Question
Integrate the functions `(sin^(-1) sqrtx - cos^(-1) sqrtx)/ (sin^(-1) sqrtx + cos^(-1) sqrtx) , x in [0,1]`
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Solution
Let I=∫sin-1x-cos-1xsin-1x+cos-1xdx
It is known that, sin-1x+cos-1x=π2
⇒I=∫π2-cos-1x-cos-1xπ2dx
=2π∫π2-2cos-1xdx
=2π.π2∫1.dx-4π∫cos-1xdx
=x-4π∫cos-1xdx ...(1)
Let I1=∫cos-1x dx
Also, let x=t⇒dx=2 t dt
⇒I1=2∫cos-1t.t dt
=2cos-1t.t22-∫-11-t2.t22dt
=t2cos-1t+∫t21-t2dt
=t2cos-1t-∫1-t2-11-t2dt
=t2cos-1t-∫1-t2dt+∫11-t2dt
=t2cos-1t-t21-t2-12sin-1t+sin-1t
=t2cos-1t-t21-t2+12sin-1t
From equation (1), we obtain
I=x-4πt2cos-1t-t21-t2+12sin-1t =x-4πxcos-1x-x21-x+12sin-1x
=x-4πxπ2-sin-1x-x-x22+12sin-1x
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