how do you integrate by

inspection/prediction.

i dont understand it, when i get stuck on an integration questions everyones like oh you need to integrate by inspection where you do trial and error. help?

Reply 1

username1765117

It’s basically just practice of seeing what thing will differentiates to give you that.

For example if I see 24x(x^2 plus 5)^5 then I immediately notice that the outside of the brackets I have a power of x that is 1 less than the inside.
This means I’ll get something similar if I differentiate (x^2 plus 5)^6.
I then try that and see that it gives me 12x(x^2 plus 5)^5 which is half what I need.
So then I can see the answer to this integral is 2(x^2 plus 5)^6 ( plus c)

(edited 5 years ago)

Reply 2

artful_lounger

Universities Forum Helper

It seems to (my impression with "by inspection" meant "I looked at it and knew the answer" lol) refer to the "art" of reverse engineering the integral, so to speak - by starting from the perspective of integration being the reverse of differentiating.

So you look at the integrand (whats inside the integral) and say "what will turn into this when I differentiate it" - then once you know what the starting function would be that differentiates to that, you know (implicitly by the fundamental theorem of calculus, for the formal terminology) that since differentiation is the opposite of integration, that function plus a constant is then the indefinite integral you're trying to find.

It might be clearer with an example;

consider int[2x]dx ; you know dy/dx (the derivative) of x^2 is 2x. Thus, the solution to the integral by inspection is x^2 + c (the constant is necessary as it's an indefinite integral).