Def integral of the product of an undef function & cosin

[crestu]

Hello,

I am looking to prove the following:

\(\displaystyle \int_{0}^{x} R(t)cos(kt) dt = \int R(x)cos(kx)dx\) for all continuous functions R(x) and all non zero constants k.

Without Tex, that is: The definate integral of R(t)cos(kt) wrt t over [0,x] is equal to the indefinate integral of R(x)cos(kx) wrt x.

(wrt = with respect to)

Any suggests would be greatly appreciated!

The full question I am trying to solve can be found at http://darkzone.cjb.cc/Robbie/Linear_Algebra/problem_question.pdf

crestu

 

[stapel]

The full question I am trying to solve...

...involves proving that a particular expression is a valid solution. In particular, that the following equation:

. . . . .\(\displaystyle \displaystyle y" + k^2y = R(x)\)

...(with k non-zero) has the following as a particular solution:

. . . . .\(\displaystyle \displaystyle y_1 = \frac{1}{k} \int_0^x R(t) \sin{(k(x - t))} dt\)

Can you plug the proposed solution into the original problem and verify that it works? Would that be sufficient "proof"?

Eliz.

 

[crestu]

Can you plug the proposed solution into the original problem and verify that it works? Would that be sufficient "proof"?

Thanks for the suggestion Eliz. That would certainly warrant proof. But wouldn't you still need to evaluate the integral in \(\displaystyle \displaystyle y= \frac{1}{k} \int_0^x R(t) \sin{(k(x - t))} dt\) in order to then be able to find the second derivative of it (wrt x)?