Question

Неопределённый интеграл!!!!

Answer 1

$int dfrac{dx}{sinx+cosx}=Big[; t=tgfrac{x}{2}; Big]=int dfrac{dt}{(1+t^2)cdot (frac{2t}{1+t^2}+frac{1-t^2}{1+t^2})}=\\\=int dfrac{dt}{-(t^2-2t-1)}=-int dfrac{d(t-1)}{(t-1)^2-2}=-dfrac{1}{2sqrt2}cdot lnBig|dfrac{t-1-sqrt2}{t-1+sqrt2}Big|+C=\\\=dfrac{1}{2sqrt2}cdot lnBig|dfrac{tgfrac{x}{2}-1-sqrt2}{tgfrac{x}{2}-1+sqrt2}Big|+C$