MCQ
$\int_{\; - \pi }^\pi {\frac{{{{\sin }^4}x}}{{{{\sin }^4}x + {{\cos }^4}x}}\;dx} = $
- A$\pi /4$
- B$\pi /2$
- C$3\pi /2$
- ✓$\pi $
$\therefore $$I = 2 \times 2\int_0^{\pi /2} {\frac{{{{\sin }^4}x}}{{{{\sin }^4}x + {{\cos }^4}x}}\;dx} $.....$(i)$
$I = 4\int_0^{\pi /2} {\frac{{{{\sin }^4}\left( {\frac{\pi }{2} - x} \right)}}{{{{\sin }^4}\left( {\frac{\pi }{2} - x} \right) + {{\cos }^4}\left( {\frac{\pi }{2} - x} \right)}}\;dx} $
$I = 4\int_0^{\pi /2} {\frac{{{{\cos }^4}x}}{{{{\cos }^4}x + {{\sin }^4}x}}\;dx} $.....$(ii)$
Adding $(i)$ and $(ii)$ we get,
$2I = 4\int_0^{\pi /2} {dx = 4 \times \frac{\pi }{2} = 2\pi } $
==> $I = \pi $.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
| Column $I$ | Column $II$ |
| $(A)$ The minimum value of $\frac{x^2+2 x+4}{x+2}$ is | $(p)$ $0$ |
| $(B)$ Let $A$ and $B$ be $3 \times 3$ matrices of real numbers, where $A$ is symmetric, $B$ is skewsymmetric, and $(A+B)(A-B)=(A-B)(A+B)$. If $(A B)^t=(-1)^k A B$, where $(A B)^t$ is the transpose of the matrix $A B$, then the possible values of $k$ are | $(q)$ $1$ |
| $(C)$ Let $\mathrm{a}=\log _3 \log _3 2$. An integer $\mathrm{k}$ satisfying $1<2^{\left(-k+3^{-2}\right)}<2$, must be less than | $(r)$ $2$ |
| $(D)$ If $\sin \theta=\cos \phi$, then the possible values of $\frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right)$ are | $(s)$ $3$ |