Solution
= Solution
The sum is finite, so <differentiation> passes through it without a convergence issue. Each lower limit $n$ is constant with respect to $x$, and the <fundamental theorem of calculus> gives the same upper-end integrand. Hence the <derivative> is
$$
\sin(x^2+x^6)\sum_{n=0}^N\binom Nn.
$$
The <binomial theorem> at $1+1$ gives the sum $2^N$. Therefore
$$
\boxed{\frac{d}{dx}\left[\sum_{n=0}^N\binom Nn
\int_n^x\sin(y^2+y^6)\,dy\right]=2^N\sin(x^2+x^6)}.
$$