My intuition for the Feynman's trick is that we construct a "morph" which produces the given function (the parameter t drives the morphing).
The key to the trick is that we construct the morph so that: a) we can tell the rate at which it increases the "area under curve" b) the rate is easier to integrate that the original function and c) the starting function has a known integral
a) is generally easier because differentiation under integral sign lets use use the standard differentiation rules.
b) this is where the difficulty in constructing the morph lies.
So we start from a known value of the integral (from c above) and then just add whatever the morph adds, which is the integral of the rate from a) over the interval of the morph.
Comments
My intuition for the Feynman's trick is that we construct a "morph" which produces the given function (the parameter t drives the morphing).
The key to the trick is that we construct the morph so that: a) we can tell the rate at which it increases the "area under curve" b) the rate is easier to integrate that the original function and c) the starting function has a known integral
a) is generally easier because differentiation under integral sign lets use use the standard differentiation rules.
b) this is where the difficulty in constructing the morph lies.
So we start from a known value of the integral (from c above) and then just add whatever the morph adds, which is the integral of the rate from a) over the interval of the morph.