luobo : i am looking forward to seeing your solution , actually , I have been shocked to see how you (or someone ) applied magic differentiation in the Integral Bee !! It is really a powerful tool
i think the solution is two straight lines passing throguh the origin (at,bt) and (-at,bt)
finding all for which the above equality holds requires considering several cases. i'm not going to do that! haha so i will only evaluate the integral.
let then:
1) therefore applying integration by parts twice gives us:
2) we know that where is the sign function.
3) using some routine trigonometry identities we get: .
4) applying 1) to 3) and then using 2) will finally give us:
The problem would be easier to solve if it changes this way:
Solve for real values of
or even
Step (1): Show for a fixed , the following integral is proportional to , i.e.
Step (2): Solve for real values of such that
For Step (1), can we use the "differentiation"? hahaha...