How to turn a limit to a definite integral

Question

I am stuck as to where to go with this problem. What I know now is that this may be a Reimann Sum problem but I am not 100% sure. I also know that the integral is lower bound by 0 and upper bound by 4.

Answer 1

Note that you can factor the expression to

$$\lim_{n\to\infty}\sum_{i=1}^n \frac{4}{n}\left(1+\left(\frac{4i}{n}\right)^2\frac{1}{16}\right)\cos\left(\frac{1}{2}\pi\left(\frac{4i}{n}\right)\right)$$

This gives you the Riemann sum $\int_0^4 (1+\frac{x^2}{16})\cos({\frac{\pi}{2} x})$.

We make use of the demonstrating Riemann sum that $\int_a^b {f(x)}{dx}=\lim_{n\to\infty}\sum_{i=1}^n{\Delta x} \cdot {f({x_i})}$, where ${\Delta x}=\dfrac{b-a}{n}$ and $x_i​=a+\Delta x\cdot i$