Find out the length of y = ln(sec x ) between 0 < x < π/4.

Solution

In this example we'll need to use the first ds as the function is in the form y = f (x). So, let us get the derivative out of the way.

 Dy/dx

= sec x tan x / secx = tan x                                          (dy/dx)² = tan² x

Let's as well get the root out of the way as there is frequently simplification that can be done and there is no cause to do that in the integral.

√ (1 + (dy/dx)²)

= √ (1+ tan² x)

= √ (sec² x )

= |sec x|

= sec x

Note: we could drop the absolute value bars here because secant is positive in the range given. After that the arc length is