Evaluate $\int{(2008 x^{2009}+2009x^{2007})}\cos(2008x) dx$

Question

$$\int{(2008 x^{2009}+2009x^{2007})}\cos(2008x) dx$$

I tried to integrate the second term by parts and also the first.None of the terms seems to cancel out.I don't know why by parts is not working.Any other method possible?

Only hints will suffice!

Answer 1

Since you only want hints:

Use integration by parts on $\displaystyle\int 2008x^{2009}\cos(2008x)\,dx$ and $\displaystyle\int 2009x^{2007}\cos(2008x)\,dx$ separately.

For the first integral, integrate the $\cos(2008x)$ term and differentiate the $x^{2009}$ term.

For the second integral, integrate the $x^{2007}$ term and differentiate the $\cos(2008x)$ term.

After you do this, what happens when you add the results together?