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

Solution 1:

Using Trigonometric substitution, $\arcsin x=\phi$

$\implies (i)x=\sin\phi$ and

$(ii)-\frac\pi2\le \phi\le\frac\pi2$ based on principal value of inverse sine ratio,

$\implies\cos\phi\ge0$

$$\int\sqrt{1-x^2}dx=\int|\cos\phi|\cos\phi d\phi = \int\cos^2\phi d\phi$$

$$= \frac12\int(1+\cos2\phi) d\phi$$

$$=\frac12\left(\phi+\frac{\sin2\phi}2\right)+K=\frac{\phi+\sin\phi\cos\phi}2+K$$

Solution 2:

$\newcommand{\cos}{\sin}$ Let's try $$\int_0^{x}\sqrt{1-a^2}\text da$$ You can generalize from that. Now, $\sqrt{1-x^2}$ describes a semicircle. Consider the following image:

So we have $\int_0^x\sqrt{1-a^2}\text da=A_1+A_2$. $A_2$ is a right triangle with base $x$ and height $\sqrt{1-x^2},$ so $A_2=\frac12x\sqrt{1-x^2}$. Meanwhile, $A_2$ is a sector of the circle. Note that the angle $$\theta=\angle (0,1)O(x,\sqrt{1-x^2})=\frac{\sin^{-1}x}1=\sin^{-1}x$$ Since the angle from the center is proportional to the area, $A_2=\frac{\theta}{2\pi}\cdot \pi r^2=\frac{\sin^{-1}x}{2}$

Thus $$ \int_0^x\sqrt{1-a^2}\text da=\frac{\sin^{-1}x+x\sqrt{1-x^2}}{2} $$