Answer:
The integral evaluates to $\sqrt{\pi}$, and the quadratic formula is used to find the roots of the quadratic equation.
Explanation:
The integral $\int_{-\infty}^{\infty} e^{-x^2} dx$ is a well-known Gaussian integral, which equals $\sqrt{\pi}$. The series expansion of the function $f(x)$ involves Fourier series components, with coefficients involving cosine and sine functions. The quadratic formula is provided for solving quadratic equations, which may relate to the roots of the quadratic part in the series expansion or in the context of the problem.
Steps:
- Evaluate the integral:
The integral $\int_{-\infty}^{\infty} e^{-x^2} dx$ is a classic integral in calculus, often solved using polar coordinates or recognizing it as a Gaussian integral.
- Known result:
- Series expansion of the function:
The function $f(x)$ is expressed as:
This is the Fourier series expansion of a periodic function with period $2L$.
- Fourier coefficients:
The coefficients $a_n$ and $b_n$ are typically calculated via integrals over one period:
- Quadratic formula:
The quadratic formula:
is used to find roots of quadratic equations, which may be relevant if the problem involves solving for specific parameters or roots related to the Fourier coefficients or the function’s properties.
Summary:
- The integral evaluates to $\sqrt{\pi}$, a standard Gaussian integral.
- The Fourier series expansion involves coefficients calculated via integrals, often involving trigonometric functions.
- The quadratic formula is a fundamental tool for solving quadratic equations, possibly used to determine parameters in the problem.
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