A curve $$'C''$$ passes through $$(2,0)$$ and the slope at $$(x,y|)$$ as $$\,{{{{\left( {x + 1} \right)}^2} + \left( {y - 3} \right)} \over {x + 3}}$$. Find the equation of the curve. Find the area bounded by curve and $$x$$-axis in fourth quadrant.

Find the value of $$\int\limits_{ - \pi /3}^{\pi /3} {{{\pi + 4{x^3}} \over {2 - \cos \left( {\left| x \right| + {\pi \over 3}} \right)}}dx} $$

If $$y\left( x \right) = \int\limits_{{x^2}/16}^{{x^2}} {{{\cos x\cos \sqrt \theta } \over {1 + {{\sin }^2}\sqrt \theta }}d\theta ,} $$ then find $${{dy} \over {dx}}$$ at $$x = \pi $$

Prove that for $$x \in \left[ {0,{\pi \over 2}} \right],$$ $$\sin x + 2x \ge {{3x\left( {x + 1} \right)} \over \pi }$$. Explain
the identity if any used in the proof.