I am reading some fairly dense engineering material and just trying to
follow some of the mathematics.  If I remembered all of my
exponentials and integration rules, this wouldn't be a problem :)  I
broke my problems reading the material in to the following two
questions.  Bonus question answers would be appreciated, but are
unnecessary.

1. Please show the step(s) that reduce

SQRT(2*E/T) 
* INTEGRAL[Bounds:0,T][COS(2*pi*f_0*t+theta) * EXP(j*2*pi*f*t) dt]

to

SQRT(T*E/2) 
* [{SIN[pi*T*(f-f_0)]/[pi*(f-f_0)]} + {SIN[pi*T*(f+f_0)]/[pi*(f+f_0)]}] 
* EXP(-j*pi*f*T) * EXP(-j*theta)

2. Similarly, please show the step(s) that reduce

INTEGRAL[Bounds:0,2T][ SIN(pi*t/2/T) * EXP(-j*2*pi*f*t) dt]

to

(4*T/pi) 
* COS(2*pi*T*f) / (1-16*T^2*f^2) 
* EXP(-j*2*pi*f*t)

I have a note that this is supposedly "easily done" using the trig
identity SIN(x) = [ EXP(jx) - EXP(-jx) ] / 2

:)


BONUS #1 (Good for a big tip):  Please show the steps that reduce

[ (T/2) * [ SIN(pi*f*T/2) / (pi*f*T/2) ] * EXP(-j*2*pi*f* T/2) ] - 
[ (T/2) * [ SIN(pi*f*T/2) / (pi*f*T/2) ] * EXP(-j*2*pi*f*3T/4) ]

to

(T/2) * [SIN(pi*f*T/2)/(pi*f*T/2)] * EXP(-j*pi*f*T) * [2*j*SIN(pi*f*T/2)]


BONUS #2 (Good for a bigger tip, if bonus #1 is answered): Please show
why EXP(-j*pi*f*T)^2 = 1.  I'm not sure if this is true, but when I
read through a couple proofs, this seems to hold.