1. Trig Integral

$\int_{0}^{ \frac{\pi}{2}} \frac{1}{(1+ \sin x)(1+ \cos x)} dx$

My first attempt was to multiply the fraction by $\frac{(1- \sin x)(1- \cos x)}{(1- \sin x)(1- \cos x)}$, but I didn't get very far with it. Not too sure on how to involve the limits usefully.

Bobak

Edit: i posted the wrong limit

2. $\int_{0}^{\pi}\frac{1}{(1+sin(x))(1+cos(x))}dx$

You can try letting $x=2tan^{-1}(u), , \;\ u=tan(x/2), \;\ du=\frac{1}{1+cos(x)}dx$

Make the subs and get:

$\int_{0}^{\infty}\frac{u^{2}+1}{u^{2}+2u+1}du$

This may look bad, but it can be expanded to:

$\int_{0}^{\infty}\left[\frac{2}{(u+1)^{2}}-\frac{2}{u+1}+1\right]du$

It is divergent.

3. Hello

Originally Posted by bobak
$\int_{0}^{ \pi} \frac{1}{(1+ \sin x)(1+ \cos x)} dx$

My first attempt was to multiply the fraction by $\frac{(1- \sin x)(1- \cos x)}{(1- \sin x)(1- \cos x)}$, but I didn't get very far with it. Not too sure on how to involve the limits usefully.

Bobak
Let me try...

We know the formulae :

$\cos x=\frac{1-t^2}{1+t^2}$ and $\sin x=\frac{2t}{1+t^2}$, where $t=\tan \frac x2$

Therefore $1+\cos x=\frac{2}{1+t^2}$ and $1+\sin x=\frac{(t+1)^2}{t^2+1}$

---> $(1+\cos x)(1+\sin x)=\frac{2(t+1)^2}{(t^2+1)^2}$

$\frac{dt}{dx}=\frac 12 \cdot (1+\tan^2 \frac x2)=\frac 12 (1+t^2) \implies dx=\frac{2dt}{1+t^2}$

If $x=0$, $\tan \frac x2=0$.

If $x \to \pi^-$, $\tan \frac x2 \to +\infty$

---> $I=\int_0^\infty \frac{t^2+1}{(t+1)^2} dt$

$I=\int_0^\infty 1-\frac{2t}{t^2+2t+1} dt$

$I=\int_0^\infty 1-\frac{2t+2}{t^2+2t+1}+\frac{2}{(t+1)^2} dt$

$I=\left[t-2\ln(t+1)-\frac{2}{t+1}\right]_0^\infty$

This is divergent...

4. Originally Posted by galactus
$\int_{0}^{\pi}\frac{1}{(1+sin(x))(1+cos(x))}dx$

You can try letting $x=2tan^{-1}(u), , \;\ u=tan(x/2), \;\ du=\frac{1}{1+cos(x)}dx$

Make the subs and get:

$\int_{0}^{\infty}\frac{u^{2}+1}{u^{2}+2u+1}du$

This may look bad, but it can be expanded to:

$\int_{0}^{\infty}\left[\frac{2}{(u+1)^{2}}-\frac{2}{u+1}+1\right]du$

It is divergent.
I posted the wrong limits but I get it now. thanks.

Bobak