How to calculate $\int\limits_{-\pi}^{\pi}f(x)\cos^5(nx)\,dx$ using Fourier series?

Question

Given Fourier series $f(x)$ in $E[-\pi,\pi]$

$f(x) \approx \frac {a_0}2 + \sum\limits_{n=1}^{\infty}(a_n\cos(nx)+b_n\sin(nx))$

Calculate $\int\limits_{-\pi}^{\pi}f(x)\cos^5(nx)\,dx$

Is there an elegant solution to this?

Answer 1

Hint. Using the angle duplication formula:

$$\cos^5 x=(\cos^2 x)^2 \cdot \cos x=\frac{(\cos 2 x+1)^2}{4} \cos x=\frac{1}{4} (\cos^2 2x+2 \cos 2x+1) \cos x=$$

$$=\frac{1}{4} \left(\frac{1}{2} (\cos 4x+1)+2 \cos 2x+1 \right) \cos x=\frac{1}{8} \cos 4x \cos x+\frac{1}{2} \cos 2x \cos x+\frac{3}{8} \cos x=$$

Now using the formula for the sum of cosines:

$$\cos 2x \cos x=\frac{1}{2} (\cos 3x+ \cos x)$$

$$\cos 4x \cos x=\frac{1}{2} (\cos 5x+ \cos 3x)$$

Finally we have:

$$\cos^5 x=\frac{1}{16} (\cos 5x+ \cos 3x)+\frac{1}{4} (\cos 3x+ \cos x)+\frac{3}{8} \cos x$$

$$\cos^5 x=\frac{1}{16} \cos 5x+\frac{5}{16} \cos 3x+\frac{5}{8} \cos x$$

Now use the definitions for the Fourier series coefficients.