Given the function $f(x) = (\pi-x)/2$ on the interval $(0,2\pi)$, with $f(0)=0$. Extend this periodically to a function over $\mathbb{R}$. The Fourier series of the function can be seen to be
$$\frac{1}{2i}\sum_{n\neq 0} \frac{e^{inx}}{n}$$
We define $S_N(f)(x) = \frac{1}{2i}\sum_\limits{n\neq 0, |n| \leq N} \frac{e^{inx}}{n}$ to be the partial sums of the above series. Then, it is easy to see that $S_N(f)(x) = \int\limits_0^x (D_n(t)-1) dt$ where $D_N(t)$ is the Dirichlet kernel defined as $D_N(t)=\frac{\sin(N+\frac{1}{2})t}{\sin t}$. Using this formulation of $S_N(f)$ as a hint, we are supposed to prove that:
$$\max\limits_{0 < x \leq \pi/N} S_N(f)(x) = \int\limits_0^{\pi}\frac{\sin t}{t}dt$$
I am having trouble trying to get a connection between the Dirichlet kernel integral and the final integral we are supposed to derive.