Find the equation of the normal to the curve
$$y = {\left( {1 + x} \right)^y} + {\sin ^{ - 1}}\left( {{{\sin }^2}x} \right)$$ at $$x=0$$

Let $$f\left( x \right) = \left\{ {\matrix{ { - {x^3} + {{\left( {{b^3} - {b^2} + b - 1} \right)} \over {\left( {{b^2} + 3b + 2} \right)}},} & {0 \le x < 1} \cr {2x - 3} & {1 \le x \le 3} \cr } } \right.$$

Find all possible real values of $$b$$ such that $$f(x)$$ has the smallest value at $$x=1$$.

The value of $$\int\limits_{\pi /4}^{3\pi /4} {{\phi \over {1 + \sin \phi }}d\phi } $$ is ..............

The value of $$\int\limits_0^{\pi /2} {{{dx} \over {1 + {{\tan }^3}\,x}}} $$ is