List of formulas that can help in calculating the integral by expanding the function into a series:

  1. Expansion of a function in a Taylor series:
    [
    f(x) = f(a) + f'(a)(x – a) + \frac{f”(a)}{2!}(x – a)^2 + \frac{f”'(a)}{3!}(x – a)^3 + \ldots
    ]
    where ( a ) is the expansion point.
  2. The integral of the Taylor series expansion:
    [
    \int f(x) \, dx = \int \left( f(a) + f'(a)(x – a) + \frac{f”(a)}{2!}(x – a)^2 + \ldots \right) dx
    ]
    Integrating each term of the series, we obtain:
    [
    = f(a)x + \frac{f'(a)}{2}(x – a)^2 + \frac{f”(a)}{3!}(x – a)^3 + \ldots + C
    ]
    where ( C ) is the constant of integration.
  3. Expansion of a function in a Fourier series (if the function is periodic):
    [
    f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos\left(\frac{2\pi nx}{T}\right) + b_n \sin\left(\frac{2\pi nx}{T}\right) \right)
    ]
    where ( a_0, a_n, b_n ) are the Fourier coefficients.
  4. Integral from a Fourier series:
    [
    \int f(x) \, dx = \int \left( a_0 + \sum_{n=1}^{\infty} \left( a_n \cos\left(\frac{2\pi nx}{T}\right) + b_n \sin\left(\frac{2\pi nx}{T}\right) \right) \right) dx
    ]
    Integrating each term yields:
    [
    = a_0 x + \sum_ {n=1}^{\infty} \left( \frac{a_n T}{2\pi n} \sin\left(\frac{2\pi nx}{T}\right) – \frac{b_n T }{2\pi n} \cos\left(\frac{2\pi nx}{T}\right) \right) + C
    ]

These formulas will help you in calculating integrals using series expansion of functions.

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