explain this

see closely this is integration by parts :D [3]

or otherwise bring the - integral on the left hand side to get \int_{a}^{b}{f d\alpha}+\int_{a}^{b}{\alpha df} = \int_{a}^{b}{ d (f\alpha)} = \left[f(x)\alpha(x) \right]_a^b

Hence proved

monotone means strictly increasing or decreasing..

I din get the use of it!

I mean what was the necessity; of it..

ony differentiable functions f and g would do!