If $f(x) = \text{Max}\{\sin x, \cos x\}$ and $g(x) = \text{Min}\{\sin x, \cos x\}$,then $\int_{0}^{\pi} f(x) dx + \int_{0}^{\pi} g(x) dx = $

Let $f: R \rightarrow R$ be a differentiable function such that its derivative $f^{\prime}$ is continuous and $f(\pi)=-6$. If $F:[0, \pi] \rightarrow R$ is defined by $F(x)=\int_0^{ x } f( t ) dt$,and if $\int_0^\pi\left(f^{\prime}( x )+ F ( x )\right) \cos x dx =2$,then the value of $f(0)$ is.

The value of the limit $\lim _{n \rightarrow \infty} \int _{0}^{1} x^{10} \sin (n x) d x$ equals

For $x, t \in R$,let $p_t(x) = (\sin t) x^2 - (2 \cos t) x + \sin t$ be a family of quadratic polynomials in $x$ with variable coefficients. Let $A(t) = \int_0^1 p_t(x) dx$. Which of the following statements are true?
$I$. $A(t) < 0$ for all $t$.
$II$. $A(t)$ has infinitely many critical points.
$III$. $A(t) = 0$ for infinitely many $t$.
$IV$. $A'(t) < 0$ for all $t$.

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 $f: R \rightarrow R$ be defined as $f(x)=a e^{2 x}+b e^x+c x$. If $f(0)=-1$,$f^{\prime}(\log _e 2)=21$,and $\int_0^{\log _e 4}(f(x)-c x) d x=\frac{39}{2}$,then the value of $|a+b+c|$ equals: