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 a function involving cosine and sine terms suggests the use of Fourier series or the general form of a function expanded in trigonometric series. The quadratic formula is a standard method for solving quadratic equations of the form $ax^2 + bx + c = 0$.
Steps:
- Recognize that the integral $\int_{-\infty}^{\infty} e^{-x^2} dx$ is a classic Gaussian integral, which is known to evaluate to $\sqrt{\pi}$.
- The series expansion:
\[
f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos \frac{n \pi x}{L} + b_n \sin \frac{n \pi x}{L} \right)
\]
is the Fourier series expansion of a periodic function over the interval $[-L, L]$.
- The quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
is used to find roots of quadratic equations, which may relate to the coefficients in the Fourier series or the parameters in the problem.
Summary:
- The integral evaluates to $\sqrt{\pi}$, a fundamental result in probability and analysis.
- The series expansion involves Fourier series concepts, decomposing functions into sinusoidal components.
- The quadratic formula is a key algebraic tool for solving quadratic equations, possibly related to the coefficients or parameters in the problem.