Spicy Limits and Integrals!

Calculus Level 4

{Ω1=∫01sin⁡−1(x1+x2)dxΩ2=∫01cos⁡−1(x1+x2)dxΩ3=lim⁡n→∞(1n+1+1n+2+…+12n)\begin{cases} \Omega_1 = \displaystyle \int_0^1 \sin^{-1} \left( \dfrac{x}{\sqrt{1+x^2}} \right) \mathrm{d}x \\ \Omega_2 = \displaystyle \int_0^1 \cos^{-1} \left( \dfrac{x}{\sqrt{1+x^2}} \right) \mathrm{d}x \\ \Omega_3 = \displaystyle \lim_{n \to \infty} \left( \dfrac{1}{n+1} + \dfrac{1}{n+2} + \ldots + \dfrac{1}{2n} \right) \end{cases}

Define Ω1,Ω2,Ω3\Omega_1, \Omega_2, \Omega_3 as of above. Then which one of the following choices is correct?

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Satyajit Mohanty by Satyajit Mohanty
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