Question

\(\frac{sin^{-1}\sqrt{x}-cos^{-1}\sqrt{x}}{sin^{-1}\sqrt{x}+cos^{-1}\sqrt{x}}\),\(x∈[0,1]\)

Answer 1

Let I=∫sin-1√x-cos-1√x/sin-1√x+cos-1√xdx

It is known that,sin-1√x+cos-1√x=π/2

⇒I=∫(π/2-cos-1√x)-cos-1√x/π/2dx

=2π/π∫(/2-2cos-1√x)dx

=2π/π./2∫1.dx4//π∫cos-1√xdx
=x-4/π∫cos-1√xdx...(1)
Let I1=∫cos-1√xdx
Also,let √x=t⇒dt=2t dt
⇒I1=2∫cos-1t.tdt
=[2cos-1t.t2/2-∫-1/√1-t2.t2/2dt]
=t2cos-1t+∫t2/√1-t2dt
=t2cos-1t-∫1-t2-1/√1-t2dt
=t2cos-1t-∫√1-t2dt+∫1/√1-t2dt
=t2cos-1t-t/2√1-t2-1/2sin-1t+sin-1t
=t2cos-1t-t/2√1-t2+1/2sin-1t
From equation(1),we obtain
I=x-4/π[t2cost-1/t√1-t2+1/2sin-1t]
=x-4/π[xcos-1√x-√x/2√1-x+1/2sin-1√x]
=x-4π/π[x(/2-sin-1√x)-√xx2/2+/2sin-1√x]
=x-2x+4x/πsin-1√x+2/π√x-x2-2/πsin-1√x
=-x+2/π[(2x-1)sin-1√x]+2/π√x-x2+C
=2(2x-1)/πsin-1√x+2/π√x-x2-x+C