I've been struggling with the integral of 1/cosx
I tried integration by parts but ended up with an even more complicated integral (xtanx/cosx)
Or use the Weierstrass sub. (Also see the cosec example above it, for step-by-step.)
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In terms of sines and cosines:
.
Now make the substitution so that and the integral becomes
Now if you make the substitution so that then the integral becomes
.
Now applying partial fractions:
and
.
So and