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[yeumontoan]

tìm nguyên hàm $\int \frac{1}{cosx+sinx+1}dx$

[nucnt772]

tìm nguyên hàm $\int \frac{1}{cosx+sinx+1}dx$

Đặt $t=tan\frac{x}{2}$ $\Rightarrow dt=\frac{1}{2}.(1+tan^{2}\frac{x}{2})dx$ $\Rightarrow dx=\frac{2dt}{1+t^{2}}$

 

$sinx=\frac{2t}{1+t^{2}}$ , $cosx=\frac{1-t^{2}}{1+t^{2}}$

 

$\int \frac{dx}{1+sinx+cosx}$$=\int \frac{2dt}{(1+t^{2}).(1+\frac{2t}{1+t^{2}}+\frac{1-t^{2}}{1+t^{2}})}$

 

 

$=\int \frac{dt}{t+1}$ $= ln|t+1|$ + C

Edited by nucnt772, 28-03-2014 - 15:04.