What is the integral of $[ (1+cos^2(x)) / sin(x) ] dx$ ?

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Answer 1 · 2016-08-06T23:38:41

$= -2 ln | csc x - cot x| + cos x + C$

$int (1+cos^2(x)) / sin(x) dx$

is just crying out for the application of the Pythagorean identity

$=int (1+ color(red)(1 - sin^2 x)) / sin(x) dx$

$=int (2 - sin^2 x) / sin(x) dx$

$=int 2 csc x - sin x dx$

these are well known integrals

$= -2 ln | csc x - cot x| + cos x + C$

What is the integral of [ (1+cos^2(x)) / sin(x) ] dx [ (1+cos^2(x)) / sin(x) ] dx?