The key for solving "hard" limits

For those who were stuck solving the integral, here are some hints:

You may want to use some trigonometric property in order to get rid of the square root. The identity

$tan^2x+1=sec^2x$

may help you rewrite the integral in this way:

$\frac{1}{2}\int_{a}^{b}sec(t)dt$

for some limits of integration. Now, this integral isn't an obvious one, hint 2:

Multiply and divide by $sec(t)+tan(t)$ .

See if you can solve it written in this way. Substitute back and get the answer!

Let,

$\displaystyle\ f(x)=\frac{1}{\sqrt{4x^{2}+9}}$ .

It's clear that $\displaystyle\ f$ is Riemann-Integrable in $\displaystyle\ [0,1]$ , hence $\displaystyle \lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=1}^{n}\frac{1}{\sqrt{4(\frac{k}{n})^{2}+9}} = \int_{0}^{1}f(x)dx = \left [\frac{1}{2}\log\left | \frac{2x}{3}+\frac{\sqrt{4x^{2}+9}}{3} \right | \right ]_{0} ^{1} \approx \boxed {0.31}$ $\square$