The area of the rectangle having vertices $\mathrm{P}, \quad \mathrm{Q}, \quad \mathrm{R}, \quad \mathrm{S}$ with position vectors $-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}},-\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$ respectively is

The value of $\int_0^1 \tan ^{-1}\left(1-x+x^2\right) \mathrm{d} x$ is

$$ \int_3^5 \frac{\sqrt{x} \mathrm{~d} x}{\sqrt{8-x}+\sqrt{x}}= $$

$$ \int \frac{\mathrm{d} x}{\sqrt{x}+x}= $$