Compute , for .
I can 'explain' how the substitution works but actually it is not the explanation , i am going to change the integral to a form which is more intuitive .
Before this , let me ask you about another integral :
Intuition or experiences or your friend tell you to subsitute so we have
so
Here is another one :
Again , they tell us to subsitute
so
The tricks of the above two integrals are very similar , what we do is to 'reflect' the integrand to obtain the answers . But we can't explain why one could think of , our intuition always cannot be explained .
This one is very similar .
Sub.
we have
Again , sub You may find that it is equivalent to sub at the begining as
Therefore ,
so
You see , these three integrals are not different at all so if you are asking how this works , let me ask you how this also works when applied in the first two integrals .