The value of $\mathrm{I}=\int_\limits{\sqrt{\log _{\mathrm{e}}}}^{\sqrt{\log _{\mathrm{e}} 3}} \frac{x \sin x^2}{\sin x^2+\sin \left(\log _{\mathrm{e}} 6-x^2\right)} d x$ is

The value of $\mathrm{I}=\int \frac{\mathrm{d} x}{x^2\left(x^4+1\right)^{\frac{3}{4}}}$ is

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three persons apply for the same house is

The value of C for which Mean value Theorem holds for the function $\mathrm{f}(x)=\log _e x$ on the interval $[1,3]$ is