If $4 \sin ^{-1} x+\cos ^{-1} x=\pi$ then $x=$

$$\int_1^e \frac{\mathrm{e}^x}{x}(1+x \log x) \mathrm{d} x=$$

The ratio of the areas bounded by the curves $y=\cos x$ and $y=\cos 2 x$ between $x=0, x=\frac{\pi}{3}$ and X -axis is

The solution of the differential equation $x \frac{\mathrm{~d}^2 y}{\mathrm{~d} x^2}=1$ at $x=y=1$ with $\frac{\mathrm{d} y}{\mathrm{~d} x}=0$ at $x=1$, is