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Evaluate the following integral:

\[\large I=\int^{3}_{1} \left( \sqrt { 1-{ (x-1) }^{ 3 } } +{ ({ x }^{ 2 }-1) }^{ \frac{1}{3} } \right) \, dx \]

\[\large \displaystyle I= \int^{1}_{0} \dfrac { \tan^{-1}(\sqrt{2+x^2})}{(1+x^2)\sqrt{2+x^2}} \,dx \]

Find the value of \(\lfloor{1000I}\rfloor\).

Notation ...

What is the exhaustive set of values of \(\displaystyle \alpha^2\) such that there exists a tangent to the ellipse \(\displaystyle x^2+\alpha^2y^2=1 \) and the portion ...

Tangent is drawn at any point \((p,q)\) on the parabola \(y^2=4ax\). Tangents are drawn from any point on this tangent to the circle ...

\[\displaystyle \prod_{ i=1 }^n \text{lcm} \left(1,2,3, \cdots, \left \lfloor \frac n i \right \rfloor \right) = k! \]

If \(n=5699\), find \(k\).

Notations:

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