Question

Неопределённый интеграл!!!!

Answer 1

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