The value of the definite integral, $\int\limits_0^{100} {\frac{x}{{{e^{{x^2}}}}}\,dx} $ is equal to

The value of integral $\int_0^1 {\frac{{{x^b} - 1}}{{\log x}}} \,dx$ is

Let $f$ be a positive function. Let

${I_1} = \int_{1 - k}^k {x\,f\left\{ {x(1 - x)} \right\}} \,dx$, ${I_2} = \int_{1 - k}^k {\,f\left\{ {x(1 - x)} \right\}} \,dx$

when $2k - 1 > 0.$ Then ${I_1}/{I_2}$ is

The points of intersection of
${F_1}(x) = \int_2^x {(2t - 5)\,dt} $ and ${F_2}(x) = \int_0^x {2t\,dt,} $ are

If $\alpha \in (2 , 3) $ then number of solution of the equation $\int\limits_0^\alpha  {}  \cos (x + \alpha^2)\, dx = \sin \,\alpha$ is :

Let $\ln x$ denote the logarithm of $x$ with respect to the base $e$. Let $S \subset R$ be the set of all points where the function $\ln \left(x^2-1\right)$ is well-defined. Then, the number of functions $f: S \rightarrow R$ that are differentiable, satisfy $f^{\prime}(x)=\ln \left(x^2-1\right)$ for all $x \in S$ and $f(2)=0$, is