Integral of e^ax·cos²(2bx) and finding all values of i^(2/3)

[Geronimo85]

I have two homework problems that have been driving me nuts:

1.) evaluate the indefinite integral:

integral(dx(e^ax)cos^2(2bx))

where a and b are real positive constants. I just don't know where to start on it.

2.) Find all values of i^(2/3)

So far I have:

i^(2/3)
= e^(2/3*ln(i))
= e^(2/3*i*(Pi/2 + 2*n*Pi))
= e^(i*Pi/3)*e^(i*n*4Pi/3)

I know from the back of my book my three solutions should end up being (1+i*sqrt(3))/2, (1-i*sqrt(3))/2, -1. But I can't seem to get there. I'd really appreciate any help. Sorry if my shorthand is confusing.

 

[Max Eilerson]

For 1) use the exponential notation for cos x, then integrate. You can do it by integration by parts twice, then use 'the trick' but that would be pretty messy.