MCQ
$\int_0^{\pi /2} {\frac{{{{\sin }^{3/2}}x\,dx}}{{{{\cos }^{3/2}}x + {{\sin }^{3/2}}x}}} = $
- A$0$
- B$\pi $
- C$\pi /2$
- ✓$\pi /4$
$= \int_0^{\pi /2} {\frac{{{{\sin }^{3/2}}\left( {\frac{\pi }{2} - x} \right)}}{{{{\cos }^{3/2}}\left( {\frac{\pi }{2} - x} \right) + {{\sin }^{3/2}}\left( {\frac{\pi }{2} - x} \right)}}dx} $
$= \int_0^{\pi /2} {\frac{{{{\cos }^{3/2}}x\,dx}}{{{{\sin }^{3/2}}x + {{\cos }^{3/2}}x}}} $.....$(ii)$
Adding $(i)$ and $(ii),$ we get $I = \frac{1}{2}\int_0^{\pi /2} {1dx = \frac{1}{2}[x]_0^{\pi /2} = \frac{\pi }{4}} $.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$S :$ Both $\sin x$ and cosx are decreasing functions in $\left( {{\pi \over 2},\pi } \right)$
$R:$ If a differentiable function decreases in $(a, b)$ then its derivative also decreases in $ (a, b).$
Which of the following is true