Problem 6.47 (a)

Integrate[(y*Cos[omega*(t+eps)]-y*Cos[omega*t])^2,{y,0,1}]

Limit[%,eps->0]

Problem 6.47 (b)

Limit[(y*Cos[omega*(t+eps)]-y*Cos[omega*t])/eps,eps->0]

Integrate[((y*Cos[omega*(t+eps)]-y*Cos[omega*t])/eps-%)^2,{y,0,1}]

Limit[%,eps->0]

Problem 6.50(a)

Integrate[(2/T)*Integrate[y*Cos[(2*Pi/T)*alpha],{alpha,t,t+T/2}],{y,0,1}]

Problem 6.50(b)

Integrate[((2/T)*Integrate[y*Cos[(2*Pi/T)*alpha],{alpha,s,s+T/2}])*((2/T)*Integrate[y*Cos[(2*Pi/T)*beta],{beta,t,t+T/2}]),{y,0,1}]

Problem 6.59

Integrate[(N0/2)*Exp[I*tau*omega]/(2*Pi),{omega,-omegab,omegab}]