Correct Answer - Option 3 :
\(\frac{2}{\sqrt{3}}\)
Given:
\(\mathop \smallint \limits_{\frac{\pi }{4}}^{\frac{\pi }{3}} \frac{{dx}}{{{{\sin }^2}x{{\cos }^2}x}}\)
Calculation:
\(\mathop \smallint \limits_{\frac{\pi }{4}}^{\frac{\pi }{3}} \frac{{dx}}{{{{\sin }^2}x{{\cos }^2}x}}\)
Multiply with 4 in the numerator and denominator
\(\mathop \smallint \nolimits_{\pi /4}^{\pi /3} \frac{{4\;dx}}{{{{\left( {2\sin x\cos x} \right)}^2}}}\)
\(\mathop \smallint \nolimits_{\pi /4}^{\pi /3} \frac{{4\;dx}}{{{{\left( {\sin \ 2x} \right)}^2}}}\)
\(4\mathop \smallint \nolimits_{\pi /4}^{\pi /3} cose{c^2}\;2x\;dx\)
\(4\left[ {\frac{{ - \cot 2x}}{2}} \right]_{\pi /4}^{\pi /3}\)
= \(\frac{{ - 4}}{2}\left[ {\cot \frac{{2\pi }}{3} - \cot \frac{{2\pi }}{4}} \right]\)
= \(-2\;\left[ {\frac{{ - 1}}{{\sqrt 3 }} - 0} \right]\)
= \(\frac{2}{{\sqrt 3 }}\)
