1. Find the arc length of the curve

Solution

The arc length is given by the integral

Make the substitution 2x = tan(t), dx = (1/2)sec2(t)dt to get

Next, integrate by parts (with u = sec(t), dv = sec2(t) dt):

That is,

or, transposing and integrating,

 

 

Finally, if q = arctan(2) then

Consequently

or