Show that c_1,c_2 Exist for Dirichlet Kernel Integral

[Zaare]

First the problem:
If $$D_n$$ is the Dirichlet kernel, I need to show that there exist positive constants $$c_1$$ and $$c_2$$ such that
$$c_1 \log n \le \int\limits_{ - \pi }^\pi {\left| {D_n \left( t \right)} \right|dt} \le c_2 \log n$$
for $$n=2,3,4,...$$.

The only thing I have been able to do is this:
$$\left| {D_n \left( t \right)} \right| = \frac{1}{\pi }\left( {\frac{1}{2} + \left| {\sum\limits_{N = 1}^n {\cos \left( {Nt} \right)} } \right|} \right) \le \frac{1}{\pi }\left( {\frac{1}{2} + n} \right)$$
Which is not good enough.
Any suggestions would be appreciated.

Edit:
By "log" I mean the natural logarithm.

 

[fresh_42]

The integrals over $\cos(Nt)|$ are all constantly $4$, hence $\int_{-\pi}^{\pi}|D_n(t)|\leq \dfrac{1}{\pi}\left(\dfrac{1}{2}+4n\right)$ by the triangle inequality. So the real problem is to find a qualitatively better approximation for $|\sum_{N=1}^{n}\cos(Nt)|$, i.e. using the negative terms in the Taylor expansions of $\cos(Nt)$.