Assertion $(A)$: $\int_0^{\frac{\pi}{2}} (\sin^6 x + \cos^6 x) dx$ lies in the interval $(\frac{\pi}{8}, \frac{\pi}{2})$.
Reason $(R)$: $\sin^6 x + \cos^6 x$ is a periodic function with period $\frac{\pi}{2}$.

Let $f: R \rightarrow R$ be a function defined as $f(x) = a \sin \left(\frac{\pi[x]}{2}\right) + [2-x]$,$a \in R$,where $[t]$ is the greatest integer less than or equal to $t$. If $\lim_{x \rightarrow -1} f(x)$ exists,then the value of $\int_{0}^{4} f(x) dx$ is equal to.

Let $F: R \rightarrow R$ be a thrice differentiable function. Suppose that $F(1)=0, F(3)=-4$ and $F'(x) < 0$ for all $x \in (1/2, 3)$. Let $f(x)=x F(x)$ for all $x \in R$.
$1.$ The correct statement$(s)$ is(are):
$(A) f'(1) < 0$
$(B) f(2) < 0$
$(C) f'(x) \neq 0$ for any $x \in (1, 3)$
$(D) f'(x)=0$ for some $x \in (1, 3)$
$2.$ If $\int_1^3 x^2 F '(x) dx = -12$ and $\int_1^3 x^3 F''(x) dx = 40$,then the correct expression$(s)$ is(are):
$(A) 9 f'(3)+f'(1)-32=0$
$(B) \int_1^3 f(x) dx = 12$
$(C) 9 f'(3)-f'(1)+32=0$
$(D) \int_1^3 f(x) dx = -12$
Give the answer for question $1$ and $2$.

Let $I_n = \int_0^1 (\log x)^n dx$,where $n$ is a non-negative integer. Then,$I_{2011} + 2011 I_{2010}$ is equal to

The greatest integer less than or equal to $\int_1^2 \log _2(x^3+1) dx + \int_1^{\log_2 9} (2^x-1)^{1/3} dx$ is . . . . .

Let $f^{\prime}(x)=\frac{192 x^3}{2+\sin ^4 \pi x}$ for all $x \in R$ with $f\left(\frac{1}{2}\right)=0$. If $m \leq \int_{1 / 2}^1 f(x) d x \leq M$,then the possible values of $m$ and $M$ are