Answer: The integral evaluates to $\sqrt{\pi}$, and the roots of the quadratic are given by the quadratic formula.
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, which are related to Fourier analysis. The quadratic formula shown is used to find the roots of a quadratic equation, which is a fundamental algebraic concept.
Steps:
- Evaluate the integral:
The integral
$$\int_{-\infty}^{\infty} e^{-x^2} dx$$
is a classic Gaussian integral.
- Method: Use polar coordinates or recognize it as a standard integral.
- Result:
$$\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$$
- Series expansion of the function:
The function
$$f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos \frac{n \pi x}{L} + b_n \sin \frac{n \pi x}{L} \right)$$
is a Fourier series expansion of a periodic function.
- Concepts involved: Fourier series, Fourier coefficients, orthogonality of sine and cosine functions.
- Quadratic formula:
The roots of the quadratic
$$ax^2 + bx + c = 0$$
are given by:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
- Concepts involved: Quadratic formula, discriminant ($b^2 - 4ac$).
Summary:
- The integral is a Gaussian integral, known to evaluate to $\sqrt{\pi}$.
- The series expansion involves Fourier series concepts.
- The quadratic formula is used to find roots of quadratic equations.