<s>[INST] Trouble solving $\int\sqrt{1-x^2} \, dx$ I am trying to learn how to solve integrals and I've got the hang out of a lot of examples, but I haven't got the slightest idea how to solve this example, this is how far I've got:
$$
\int\sqrt{1 - x^2} \, dx = x\sqrt{1-x^2} - 2\int\frac{x^2}{\sqrt{1-x^2}} \, dx
$$
Can you please help me solve it, and also some tips concerning the integration are welcome.
Thank you
 [/INST] Your idea of integrating by parts is good (but you did it wrong):
\begin{align}
\int\sqrt{1 - x^2} \, dx 
&= x\sqrt{1-x^2} - \int x\frac{-2x}{2\sqrt{1-x^2}} \, dx \\
&= x\sqrt{1-x^2} + \int\frac{x^2}{\sqrt{1-x^2}}\,dx\\
&= x\sqrt{1-x^2} + \int\frac{x^2-1+1}{\sqrt{1-x^2}}\,dx\\
&= x\sqrt{1-x^2} - \int\sqrt{1-x^2}\,dx+\int\frac{1}{\sqrt{1-x^2}}\,dx\\
&= x\sqrt{1-x^2} + \arcsin x - \int\sqrt{1-x^2}\,dx
\end{align}
Therefore
$$
2\int\sqrt{1 - x^2} \, dx = x\sqrt{1-x^2} + \arcsin x
$$
and
$$
\int\sqrt{1 - x^2} \, dx = \frac{1}{2}x\sqrt{1-x^2} + \frac{1}{2}\arcsin x + c
$$
</s>