How do you integrate $(1-cosx)/(1+cosx)$ ?

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Answer 1 · 2015-04-17T18:18:03

First just do simple math :

$int(1-cos(x))/(1+cos(x))dx = (-1-cos(x)+2)/(1+cos(x))dx$

Now we can factorize the numerator :

$=>int(-(1+cos(x))+2)/(1+cos(x))dx$

$=>int(-(1+cos(x)))/(1+cos(x))dx+2int1/(1+cos(x))dx$

$=>int -1dx+2int1/(1+cos(x))dx$

Remember that $cos^2(x) = 1/2(1+cos(2x))$

(From $cos(a)cos(b)$ formula with $a = b$ )

$=>2cos^2(x) = (1+cos(2x))$

$=>2cos^2(1/2x) = 1+cos(x)$

$=>int -1dx+2int1/2*1/cos^2(1/2x)dx$

$1/2*1/cos^2(1/2x)dx$ is exactly the derivate of $tan(1/2x)$

$=>[2tan(1/2x) - x] + C$

How do you integrate (1-cosx)/(1+cosx)(1-cosx)/(1+cosx)?