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

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
$$
\int\sqrt{1 - x^2} \, dx = \frac{1}{2}x\sqrt{1-x^2} + \frac{1}{2}\arcsin x + c
$$
