Integrate with substitution Use the substitution $x=2\sin(\theta)$ to find the value of
$$\int_0^{\sqrt{3}} \frac{1}{(4+x^2)^{3/2}}dx$$
I got to $$\int_{0}^{\pi/3} \frac{2\cos(\theta)}{(4+4\cos^2(\theta))^{3/2}}d\theta.$$
However, I don't know how to further integrate this!
Thanks!

For now let's perform the indefinite integral: $$\int \frac{1}{(4+x^2)^{3/2}}dx.$$
If we use the substitution $x=2\tan(\theta)$ this gives $dx=2\sec^2(\theta)d\theta$.
Plugging in the subsitution yields: $$\int \frac{2\sec^2(\theta)}{(4+4\tan^2(\theta))^{3/2}} d\theta=\frac{1}{4} \int \frac{\sec^2(\theta)}{\sec^3(\theta)} d\theta =\frac{1}{4} \int \frac{1}{\sec(\theta)}d\theta =\frac{1}{4} \int \cos(\theta)d\theta =\frac{1}{4} \sin(\theta)+C.$$
Since $x/2 = \tan(\theta)$, this means $$\sin(\theta) = \frac{x}{\sqrt{x^2+4}}.$$
