Spicy Limits and Integrals!

Calculus Level 5

\[\begin{cases} \Omega_1 = \displaystyle \int_0^1 \sin^{-1} \left( \dfrac{x}{\sqrt{1+x^2}} \right) \mathrm{d}x \\ \Omega_2 = \displaystyle \int_0^1 \cos^{-1} \left( \dfrac{x}{\sqrt{1+x^2}} \right) \mathrm{d}x \\ \Omega_3 = \displaystyle \lim_{n \to \infty} \left( \dfrac{1}{n+1} + \dfrac{1}{n+2} + \ldots + \dfrac{1}{2n} \right) \end{cases} \]

Define \(\Omega_1, \Omega_2, \Omega_3\) as of above. Then which one of the following choices is correct?

\[\Omega_1 + \Omega_2 + \Omega_3 = 4\] \[\Omega_1 + \Omega_2 + \Omega_3 = \ln(2)\] \[\Omega_1 + \Omega_2 = \Omega_3\] \[\Omega_1 + \Omega_2 + \Omega_3 = 0\] \[\Omega_1 + \Omega_3 = \Omega_2\] \[\Omega_2 + \Omega_3 = \Omega_1\] \[\Omega_1 + \Omega_2 + \Omega_3 = 1\] \[\Omega_1 + \Omega_2 + \Omega_3 = \ln(4)\]

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