Why does integration by parts work?

Recall the product rule for differentiation: the derivative of uv is equal to u'v+uv'.If we use the fact that integration reverses differentiation (so the integral of f' is f), then we calculate that uv is equal to integral of u'v+uv'. We can then rearrange this to get that the integral of u'v is equal to uv minus the integral of uv'.The reason integration by parts is useful is that if we may not know how to integrate u'v, but if we do know how to integrate uv', we can find the solution. A good example is how should we integrate x cos(x)?Lets choose u'=cos(x), v=x. Then we know that u=sin(x), v'=1.So the integral of x cos(x) is equal to x sin(x) minus the integral of sin(x)*1=sin(x). Hence the integral of x cos(x) is equal to x sin(x) +cos(x).