A sign -post in the form of an isosceles triangle $$ABC$$ is mounted on a pole of height $$h$$ fixed to the ground. The base $$BC$$ of the triangle is parallel to the ground. A man standing on the ground at a distance $$d$$ from the sign-post finds that the top vertex $$A$$ of the triangle subtends an angle $$\beta $$ and either of the other two vertices subtends the same angle $$\alpha $$ at his feet. Find the area of the triangle.

Investigate for maxima and minimum the function $$$f\left( x \right) = \int\limits_1^x {\left[ {2\left( {t - 1} \right){{\left( {t - 2} \right)}^3} + 3{{\left( {t - 1} \right)}^2}{{\left( {t - 2} \right)}^2}} \right]} dt$$$

The integral $$\int\limits_0^{1.5} {\left[ {{x^2}} \right]dx,} $$

Where [ ] denotes the greatest integer function, equals .............

The value of the integral $$\int\limits_0^{2a} {[{{f\left( x \right)} \over {\left\{ {f\left( x \right) + f\left( {2a - x} \right)} \right\}}}]\,dx} $$ is equal to $$a$$.