Question

${\int }_0^{\frac{\pi }{2}}\frac{si{n}^{\frac{3x}{2}}dx}{co{s}^{\frac{3x}{2}}+si{n}^{\frac{3x}{2}}}=$ [Roorkee 1989; BIT Ranchi 1989]

• A. $0$
• B.
  
• C.
• D.
  

Answer 1

The correct option is D

$LetI={\int }_0^{\frac{\pi }{2}}\frac{si{n}^{\frac{3x}{2}}dx}{co{s}^{\frac{3x}{2}}+si{n}^{\frac{3x}{2}}}\cdots \left(i\right)$
$={\int }_0^{\frac{\pi }{2}}\frac{si{n}^{\frac{3}{2}}(\frac{\pi }{2}-x)}{co{s}^{\frac{3}{2}}(\frac{\pi }{2}-x)+si{n}^{\frac{3}{2}}(\frac{\pi }{2}-x)}dx$
$={\int }_0^{\frac{\pi }{2}}\frac{co{s}^{\frac{3x}{2}}dx}{si{n}^{\frac{3x}{2}}+co{s}^{\frac{3x}{2}}}\cdots \left(ii\right)$
Adding (i) and (ii), we get $I=\frac{1}{2}{\int }_0^{\frac{\pi }{2}}1dx=\frac{1}{2}[x{]}_0^{\frac{\pi }{2}}=\frac{\pi }{4}$