How to integrate this? quick question

[NutriGrainKiller]

$$\int_{0}^{2 \lambda} \cos({\frac{kx}{2}}) \cos({nx}) dx$$

I can't find my calc2 notes and it's killing me! This thing came up half way through the computation of the Fourier series of $$f(x) = A\cos({\frac{\pi x}{\lambda}})$$..and i can't remember how to do it!

I am very aware of the trig identity below, but i would prefer not to use it for obvious reasons.

$$\cos{\alpha}\cos{\beta} = \frac{1}{2}[\cos{\frac{(\alpha + \beta)}{2} + \cos{\frac{(\alpha - \beta)}{2}]$$

I have the exact same question except instead of two cosines there's one cosine and one sine..any help is appreciated!

 

[HallsofIvy]

What are the obvious reasons? I would think that using that identity would be the "obvious" way to do it!

(Your identity is wrong: there is no 1/2 inside the functions. It is
$$\cos{\alpha}\cos{\beta} = \frac{1}{2}[\cos(\alpha + \beta) + \cos{(\alpha - \beta)]$$)

And
$$sin(\alpha)cos(\beta)= \frac{1}{2}[sin(\alpha+ \beta)+ sin(\alpha-\beta)]$$