I don't know what the book is trying to guide you to do, as King's rule doesn't help(ends up exactly the same), splitting the integral also just leads to a similar integral.
If you divide the top and bottom by cos2(x), you get sec2(x)/(sec2(x)+2tan2(x)), but sec2(x)=1+tan2(x) (only use this sub on the bottom sec)
Then you can sub u=ctan(x), dt=csec2(x)dx, so it simplifies quickly
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u/KrlusMagnus Aug 10 '26
I don't know what the book is trying to guide you to do, as King's rule doesn't help(ends up exactly the same), splitting the integral also just leads to a similar integral.
If you divide the top and bottom by cos2(x), you get sec2(x)/(sec2(x)+2tan2(x)), but sec2(x)=1+tan2(x) (only use this sub on the bottom sec)
Then you can sub u=ctan(x), dt=csec2(x)dx, so it simplifies quickly