Correct Answer - Option 4 : π/4
Concept:
Properties of definite integral:
\(\rm \int\limits_a^b f(x) dx\) = \(\rm \int\limits_a^b f(b+a -x) dx\)
Calculation:
I = \(\rm\int\limits_0^{\pi /2} {\frac{{{{\cos}^{1000}}x}}{{{{\sin }^{1000}}x + {{\cos }^{1000}}x}}}dx \) ...(i)
I = \(\rm\int\limits_0^{\pi /2} {\frac{{{{\cos}^{1000}}({\pi\over2}+0-x)}}{{{{\sin }^{1000}}({\pi\over2}+0-x)+ {{\cos }^{1000}}({\pi\over2}+0-x)}}}dx \)
I = \(\rm\int\limits_0^{\pi /2} {\frac{{{{\sin}^{1000}}(x)}}{{{{\cos}^{1000}}(x)+ {{\sin}^{1000}}(x)}}}dx \) ....(ii)
Adding (i) and (ii), we get
2I = \(\rm\int\limits_0^{\pi /2} {\frac{{{{\cos}^{1000}}x + {{{\sin}^{1000}}x}}}{{{{\sin }^{1000}}x + {{\cos }^{1000}}x}}}dx \)
2I = \(\rm\int\limits_0^{\pi /2} dx \)
2I = \(\rm[x]_0^{\pi /2}\)
2I = \({\pi\over2}-0\)
I = \(\boldsymbol{\pi\over4}\)
