laplace transform is very useful for finding various parameters and solutions to complex circuits and equations.
u can get some idea abt it from here. http://en.wikipedia.org/wiki/Laplace_Transform
u may for now have to solve the above equation using solution methods for differential equations.
4*I + (8*I)/(j*w) - 3*j*w*I = 50<75
w = omega = 2
4*I - (j*8*I)/2 - j*6*I = 50<75
add the imaginary terms
4*I - j*65*I = 50<75
convert to polar form
65.123<-86.5 * I = 50<75
divide the magnitudes and subtract the angles
I = 0.7678<161.5
i = 0.7678*cos(2t + 161.5 degrees) A
This is very similar to the Laplace transform...basically "s" becomes "jw", like Fourier. But w is constant, so it is in "phase-domain" instead of frequency-domain.
You might want to double-check my algebra/arithmetic, I'm not really that awake.