Question

I have seen some people substitute in a simpler way for example: Integral sin^2xcosxdx
they write: integral sin^2xd(sinx)
Well I get it for the easier example above but when comes to tougher substitution I don't quite get it. SO there must be a way to conceptualize this trick. Anyone with help?

Answer 1

try using u substitution idea; so if u=sin(x) then du=cos(x). Therefore, we get the integral of u^2du. Does that make more sense?

Answer 2

I always substitute U but some of my my Chinese friends just skip few steps and directly do it!

Answer 3

ya, they have probably done those integrals (or seen some like it) so many times that they can do it mentally; some of my teachers can do that too. Happens over time

Answer 4

Well, the best way to be able to mentally solve these equations is by identifying some standard derivatives in your integrand. In the example you gave above once you see that cos(x) is a derivative of sin(x) i.e d sin(x) /d x = cos(x) you re-arrange to get d (sin(x)) = cos(x) dx. Now some times this is not very useful so you need to see if the process of replacing cos(x) dx with d (sin(x)) actually helps which in the above case makes it simpler.

Hope this helps :)