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How to do this integration: $\int \sec (x-a) \sec (x-b)\ dx?$

I want to this in the shortest possible way. Please guide me through.

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    Probably not the _shortest possible way_ but you can try to: 1. Type "integrate 1/[cos(x-a)*cos(x-b)]" into wolframalpha; 2. wait for a while; 3. click on "show steps" http://www.wolframalpha.com/input/?i=integrate+1%2F%5Bcos%28x-a%29%2Acos%28x-b%29%5D2011-12-08

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This can be done via the following trick. Rewrite your integral as ${1 \over \sin(b - a)} \int {\sin(b - a) \over \cos(x - a)\cos(x - b)}\,dx$ Note that $b - a = (x - a) - (x -b)$, so by the sine subtraction formula you have $\sin(b - a) = \sin(x - a)\cos(x - b) - \sin(x - b)\cos(x - a)$ Subtituting this back into your integral it becomes ${1 \over \sin(b - a)} \bigg(\int \tan(x - a)\,dx- \int \tan( x - b)\,dx\bigg)$ $ = {1 \over \sin(b - a)} \bigg(-\ln(\cos(x - a)) + \ln(\cos(x - b))\bigg) + C$

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    Yeah,this is actually a text book problem :)2011-12-08