$$\int\frac{1}{(x^2-1)\sqrt{x^2+1}}dx$$
I'm trying to solve this integral. First I substituted : $x=\tan(t)$; $t=\arctan(x)$
Then $$ dx=\frac{1}{\cos^2(t)}\,dt$$
Now by simplifying I'm to this step $$ \int\frac{\cos(t)}{\sin^2(t)-\cos^2(t)}\,dt$$
What can i do now ..
Thank you in advance 🙂
Best Answer
$$\int\frac{\cos t}{\sin^2 t-\cos^2 t}dt$$
Use $\cos^2 t = 1-\sin^2 t$ to modify the denominator $$\int\frac{\cos t}{2\sin^2 t-1}dt$$ Substitute $u=\sin t$, $du =\cos t \, dt$ $$\int\frac{1}{2u^2-1}du$$ Now integrate using partial fractions or otherwise.