Let $$f\left( x \right)$$  be a continuous function such that the area bounded by the curve $$y = f\left( x \right),\,\,x$$   -axis and the lines $$x = 0$$  and $$x = a$$  is $$\frac{{{a^2}}}{2} + \frac{a}{2}\sin \,a + \frac{\pi }{2}\cos \,a,$$       then $$f\left( {\frac{\pi }{2}} \right) = ?$$

Solution :
We have, $$\int\limits_0^a {f\left( x \right)dx} = \frac{{{a^2}}}{2} + \frac{a}{2}\sin \,a + \frac{\pi }{2}\cos \,a$$
Differentiating w.r.t. $$a,$$ we get
$$\eqalign{ & f\left( a \right) = a + \frac{1}{2}\left( {\sin \,a + a\,\cos \,a} \right) - \frac{\pi }{2}\sin \,a \cr & {\text{Put }}a = \frac{\pi }{2}\,; \cr & f\left( {\frac{\pi }{2}} \right) = \frac{\pi }{2} + \frac{1}{2} - \frac{\pi }{2} = \frac{1}{2} \cr} $$