I did it by the method of integration by parts, with
$$
u=\frac{dx}{\cos(x) + \sin(x)},\quad dv=dx$$
so
$$
\int \frac{dx}{\cos(x) + \sin(x)} =
\frac{x}{\cos(x)+\sin(x)}
– \int \frac{x(\sin(x)-\cos(x))}{(\cos(x)
+ \sin(x))^{2}}
$$
Where,
$$\int \frac{x(\sin(x)-\cos(x))}{(\cos(x) + \sin(x))^{2}}\;dx
= \int \frac{x \sin(x)}{1+2\sin(x)\cos(x)}dx – \int \frac{x \cos(x)}{1+2\sin(x)\cos(x)}dx ,
$$
I have not managed to solve those two integrals that were expressed, really appreciate if you can help me.
Best Answer
Use$$\int\frac{dx}{\cos x+\sin x}=\int\frac{1}{\sqrt{2}}\sec(x-\pi/4)dx=\frac{1}{\sqrt{2}}\ln|\sec(x-\pi/4)+\tan(x-\pi/4)|+C.$$