If $\int_{}^{} {f(x)\,dx} = x{e^{ - \log |x|}} + f(x),$ then $f(x)$ is

Let $f:[0,1] \rightarrow[0, \infty)$ be a continuous function such that $\int_0^1 f(x) d x=10$. Which of the following statements is NOT necessarily true?

Number of values of $x$ satisfying the equation

$\int\limits_{ - \,1}^x {\,\left( {8{t^2} + \frac{{28}}{3}t + 4} \right)\,dt} $ $=$ $\frac{{\left( {{\textstyle{3 \over 2}}} \right)x + 1}}{{{{\log }_{(x + 1)}}\sqrt {x + 1} }}$ , is

Let $\operatorname{Max} \limits _{0 \leq x \leq 2}\left\{\frac{9-x^{2}}{5-x}\right\}=\alpha$ and $\operatorname{Min} \limits _ {0 \leq x \leq 2}\left\{\frac{9-x^{2}}{5-x}\right\}=\beta$

If $\int\limits_{\beta-\frac{8}{3}}^{2 a-1} \operatorname{Max}\left\{\frac{9- x ^{2}}{5- x }, x \right\} dx =\alpha_{1}+\alpha_{2} \log _{e}\left(\frac{8}{15}\right)$ then $\alpha_{1}+\alpha_{2}$ is equal to

Let $f(x)=2+|x|-|x-1|+|x+1|, x \in R$. Consider

$(S1)$: $f^{\prime}\left(-\frac{3}{2}\right)+f^{\prime}\left(-\frac{1}{2}\right)+f^{\prime}\left(\frac{1}{2}\right)+f^{\prime}\left(\frac{3}{2}\right)=2$

$( S 2): \int_{-2}^{2} f ( x ) dx =12$Then,

The true set of values of $‘a’$ for which the inequality $\int\limits_a^0 {} (3^{ -2x} - 2. 3^{-x})\, dx \geq 0$ is true is: