Let $f(x) = \min \{[x-1], [x-2], \ldots, [x-10]\}$ where $[t]$ denotes the greatest integer $\leq t$. Then $\int_{0}^{10} f(x) \, dx + \int_{0}^{10} (f(x))^2 \, dx + \int_{0}^{10} |f(x)| \, dx$ is equal to

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 $t$.

If $I = \sum_{k=1}^{98} \int_k^{k+1} \frac{k+1}{x(x+1)} dx$,then which of the following is true?

The numbers $P, Q$ and $R$ for which the function $f(x) = P{e^{2x}} + Q{e^x} + Rx$ satisfies the conditions $f(0) = -1$,$f'(\log 2) = 31$ and $\int_0^{\log 4} [f(x) - Rx] \, dx = \frac{39}{2}$ are given by

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.

An extremum value of $y = \int_{0}^{x} (t - 1)(t - 2) dt$ is