What is the definite integral of (sec^4 x) from 0 to pi/4? Just for clarificatio

Question

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 on a quiz, and I "saved" a sec^2x and then made the other sec^2x equal to (tan^2x + 1) and then let u = tanx. I was able to solve that indefinitely, but the problem comes about because tan(0) is undefined and that is the lower limit of integration. Any help would be much appreciated--This problem is going to bothering me all day until I get my quiz back! :)

Explanation / Answer

(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+.

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