Care to share what you've tried? For reference, Simpson's Rule (and in particular, Composite Simpson's Rule).
Set up your subintervals and then just apply the formula.
People are usually more willing to help if you demonstrate that you've made an effort to solve the problem. An easy way to do this is to show any work you've done. And if you can't figure out where to start, you should give specifics on what you don't understand about the process.
I'll set it up this time, though. We want to split [0, 5] into 10 subintervals. That means each subinterval would have to be $\displaystyle \frac{5-0}{10}=\frac12$ units wide. Therefore, the values separating the subintervals are
$\displaystyle \begin{array}{ccc}t_0&=&0 \\ t_1&=&\frac12 \\ t_2&=&1 \\ t_3&=&\frac32 \\ \vdots&&\vdots \\ t_n&=&\frac n2\end{array}$
Now we put these values into the formula.
$\displaystyle \int_0^5t\sqrt{7+8t^2}\,dt$
$\displaystyle =\int_a^b f(t)\,dt$
$\displaystyle \approx\frac{b-a}{3n}\left[f(t_0)+4f(t_1)+2f(t_2)+4f(t_3)+\dots+2f(t_8)+4f(t_ 9)+f(t_{10})\right]$
$\displaystyle =\frac16\left[f(0)+4f\left(\frac12\right)+2f(1)+4f\left(\frac32 \right)+\dots+2f(4)+4f\left(\frac92\right)+f(5) \right]$
Now evaluate the integrand (which I'm calling $\displaystyle f(t)$) at each endpoint.
First of all here is a link to the Simpson's Rule with shown examples.
Secondly this integral can be solved directly using substitution rule.
Let $\displaystyle u=7+8t^2$ then du=16tdt and $\displaystyle tdt=\frac{du}{16}$.
Since x is changing from 0 to 5 then u is changing from 7 to 207 and your integral now is $\displaystyle \frac{1}{16}\int_7^{207}\sqrt{u}du=\frac{1}{16} \frac23u^{\frac32}|_7^{207}=\frac{1}{24}({207}^{ \frac32}-7^{\frac32})$
Now, if you are still don't understand Simpson's rule than divide interval [0,5] on n=10 subintervals with length $\displaystyle Dx=\frac{5-0}{10}=0.5$
So, $\displaystyle t_i=0+Dx*i$: $\displaystyle t_0=0, t_1=0.5,t_2=1, t_3=1.5,...,t_{10}=1$
Your function is $\displaystyle f(x)=t\sqrt{7+8t^2}$
Therefore, by Simpson's rule approximation is $\displaystyle \frac{Dx}{3}(f(0)+4f(0.5)+2f(1)+4f(1.5)+2f(2)+4f(2 .5)+2f(3)+4f(3.5)+2f(4)+4f(4.5)+f(5))$
Plug values and calculate. For example $\displaystyle f(0)=0*\sqrt{7+8*0^2}=0$