Sunday, August 13, 2017

trigonometry - How to simplify Releft[sqrt2tan1xoversqrtiright]?



While solving x2+1x4+1dx, I tried to use partial fractions in the denominator by writing x4+1=(x2+i)(x2i) And then I got [2tan1xi]. In my book they used another method without complex numbers and they got 12tan1(x21x2). How do I prove my answer is equal to theirs? I tried the realtions between log and atan but I couldnt get rid of the i. Edit: Please note that this question is not about solving the integral (which I already solved to get [2tan1xi].) but about the simplification of the answer I got using complex partial fractions method to reduce it to the real part only.


Answer



[2tan1xi]=[2tan1xeiπ/4]=(2tan1xeiπ/4)+(2tan1xeiπ/4)2


=12(tan1xeiπ/4+tan1xeiπ/4)=12tan1xeiπ/4+xeiπ/41xeiπ/4xeiπ/4


=12tan12xcosπ41x2=12tan12x1x2


where line 23 is achieved by using that tan(x+y)=tanx+tany1tanxtany.


The differs from the expression you give by a constant, after further noting that tan1x+tan1(1x)=π2 for x>0.


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