Integrate by parts to find some useful recurrences relating Iₙ and Jₙ.
[tex]\displaystyle I_n = \int_{-\pi}^\pi x^n \cos(x) \, dx = uv \bigg|_{x=-\pi}^{x=\pi} - \int_{-\pi}^\pi v\, du \\\\ \implies I_n = -n \int_{-\pi}^\pi x^{n-1} \sin(x) \, dx \\\\ \implies I_n = -n J_{n-1}[/tex]
where u = xⁿ and dv = cos(x) dx.
[tex]\displaystyle J_n = \int_{-\pi}^\pi x^n \sin(x) \, dx = uv \bigg|_{x=-\pi}^{x=\pi} + \int_{-\pi}^\pi v\, du \\\\ \implies J_n = \pi^n - (-\pi)^n + n \int_{-\pi}^\pi x^{n-1} \cos(x) \, dx \\\\ \implies J_n = \pi^n - (-\pi)^n + n I_{n-1}[/tex]
The integrals for n = 0 are trivial:
[tex]\displaystyle I_0 = \int_{-\pi}^\pi \cos(x) \, dx = \sin(\pi) - \sin(-\pi) = 0[/tex]
[tex]\displaystyle J_0 = \int_{-\pi}^\pi \sin(x) \, dx = -\cos(\pi) - (-\cos(-\pi)) = 0[/tex]
Now, the integral we're interested in is
[tex]\displaystyle \int_{-\pi}^\pi x^n f(x) \cos(x) \, dx[/tex]
but we know f(x) is quadratic, and we want to find its coefficients a, b, and c such that
[tex]\displaystyle \int_{-\pi}^\pi x^n (ax^2+bx+c) \cos(x) \, dx[/tex]
But this is simply
[tex]\displaystyle \int_{-\pi}^\pi (ax^{n+2}+bx^{n+1}+cx^n) \cos(x) \, dx = aI_{n+2} + bI_{n+1} + cI_n[/tex]
Using the recurrences above and the initial values we've computed, we find
• I₁ = -J₀ = 0
• J₁ = π - (-π) + I₀ = 2π
• I₂ = -2 J₁ = -4π
• J₂ = π² - (-π)² + 2 I₁ = 0
• I₃ = -3 J₂ = 0
• J₃ = π³ - (-π)³ + 3 I₂ = 2π³ - 12π
• I₄ = -4 J₃ = -8π³ + 48π
When n = 0, the integral we care about is
[tex]aI_2 + bI_1 + cI_0 = -4\pi a + 0 + 0 = 4\pi \implies a = -1[/tex]
When n = 1,
[tex]aI_3 + bI_2 + cI_1 = 0 - 4\pi b + 0 = 4\pi \implies b = -1[/tex]
When n = 2,
[tex]aI_4 + bI_3 + cI_2 = (48\pi - 8\pi^3)a + 0 - 4\pi c = 4\pi \implies c = 2\pi^2 - 13[/tex]
so that
[tex]f(x) = \boxed{-x^2 - x + 2\pi^2 - 13}[/tex]
