At my university, some students organized an unofficial integral race. I participated in the final round and emerged as the winner, successfully solving two integrals while my opponent managed to solve only one integral. This is integral number 6
\begin{align} \int \frac{\text{sinc}\left(\sqrt{e^x}\right)}{4a^2} + x^{\phi(p)} \, dx \end{align}
Where $\phi(p)$ is totient function
I have tried but it doesn't work.how do i evaluate it any advice,hint or well thought solution will be appreciated
Rewrite the integrand and solve
$$ \int \frac{\sin \sqrt{e^x}}{2a^2 e^x}\cdot \frac{\sqrt{e^x}}{2} + x^{\phi(p)} \: dx = \frac{x^{\phi(p)+1}}{\phi(p)+1}-\frac{\sin\sqrt{e^x}}{2a^2\sqrt{e^x}}+\int\frac{\cos\sqrt{e^x}}{2a^2\sqrt{e^x}}\cdot\frac{\sqrt{e^x}}{2}\:dx$$
$$= \boxed{\frac{x^{\phi(p)+1}}{\phi(p)+1}-\frac{\operatorname{sinc}\sqrt{e^x}}{2a^2}+\frac{\operatorname{Ci}\sqrt{e^x}}{2a^2}+C}$$
by integration by parts