What is the definite integral of (sec^4 x) from 0 to pi/4? Just for clarification, that is a secant to the fourth power of x. I had this question...

(secx)^4  = (1+tan^2 x) sec^2

Therefore Integral (sec^4 x) dx = Integral (1+tan^2 x) sec^2 x dx

Let tan x=t , then sec^2 x dx = dt , for x=0, t=0  and for x=pi/4 , t=1, with this transformation,

Right side is  integral (1+t^2) dt

=(t+t^3/3)+C . Taking  itegral limits from t= 0 to t=1 , we get

=(1+1^3/3+C)- (0+0+C)

=4/3

NB : Tan 0 = 0 and it is not undefined, whereas tan 90 = is undefined with change of sign and infinite jump from  +inf to -inf as go from 90-  to  90+.