<s>[INST] How to proceed with a definite integral of this form? How do I find the definite integral of this sort of function?
$$ \int_m^n \frac{\sin(a  \sin(\sqrt{(b  \lfloor x\rfloor^2 + b  d + 1)}))}{a \sin(\sqrt{(b  \lfloor x\rfloor^2 + b  d + 1)})} dx  $$
This function is actually a form of the $ \operatorname{sinc} $ function except it has a floor variable nested inside it, and also takes the sine of another sine function.
Note: a & b = 10^k (a & b are higher order powers of 10)
 [/INST] \begin{eqnarray}
\int_m^n \frac{\sin(a  \sin(\sqrt{(b  \lfloor x\rfloor^2 + b  d + 1)}))}{a \sin(\sqrt{(b  \lfloor x\rfloor^2 + b  d + 1)})} dx&=&\sum_{k=m}^n\int_k^{k+1} \frac{\sin(a  \sin(\sqrt{(b  k^2 + b  d + 1)}))}{a \sin(\sqrt{(b k^2 + b  d + 1)})} dx\\
&=&\sum_{k=m}^n \frac{\sin(a  \sin(\sqrt{(b  k^2 + b  d + 1)}))}{a \sin(\sqrt{(b k^2 + b  d + 1)})}
\end{eqnarray}
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