Solving Integral Problem: Limit of Sum of Cosines

[Upiór]

I have a problem with this integral. By the way, it's not my homework:)

\int\limits_{0}^{+\infty}(\lim\limits_{n\rightarrow+\infty} \displaystyle\frac{1+\cos\frac{x}{n}+\cos\frac{2x}{n}+\ldots+\cos \frac{(n-1)x}{n}}{n} \right))dx

 

[Count Iblis]

The limit inside the integrand can be written as an integral. If you compute that integral, you find that it is equal to sin(x)/x. If you integrate that from zero to infinity, you get pi/2.

 

[Upiór]

Hm, but how to compute this integral inside? Why is it equal to sinx/x? I cannot see the way...

Step by step, please:)