# Integral of Legendre polynomial

• shaun_chou
In summary, the integral of Legendre polynomial is a mathematical operation used to calculate the area under the curve of a Legendre polynomial function. It has various applications in physics, engineering, and other fields of mathematics, such as solving differential equations and calculating moments of inertia. It can be solved analytically, but for higher order polynomials, numerical methods may be necessary. The integral of Legendre polynomial is also related to Legendre-Gauss quadrature, a numerical method for approximating integrals, and has real-world applications in fields such as mechanical engineering, electrostatics, and signal processing.
shaun_chou

## Homework Statement

In solving a question I got a problem of solving the following integral. Your comments are appreciated.

## Homework Equations

$$\int_{-1}^{1}xP_l'(x)dx=?$$

## The Attempt at a Solution

I tried to solve by integration by parts, i.e.
$$\left{}xP_l(x)\right{|}_{-1}^{1}-\int_{-1}^{1}P_l(x)$$ but I can't get the simple solution to $$\int_{-1}^{1}P_l(x)$$

Hint: the Legendre polynomials satisfy an orthogonality relationship...

Thanks a lot for your help.

## 1. What is the definition of the integral of Legendre polynomial?

The integral of Legendre polynomial is a mathematical operation that calculates the area under the curve of a Legendre polynomial function. It is represented by the symbol ∫P(x)dx, where P(x) is the Legendre polynomial and dx represents the differential element.

## 2. What is the significance of the integral of Legendre polynomial in mathematics?

The integral of Legendre polynomial has various applications in physics, engineering, and other fields of mathematics. It is used to solve differential equations, calculate moments of inertia, and find the probability distribution of certain physical systems.

## 3. Can the integral of Legendre polynomial be solved analytically?

Yes, the integral of Legendre polynomial can be solved analytically using integration techniques such as substitution, integration by parts, or partial fractions. However, for higher order Legendre polynomials, the integration becomes more complex and may require numerical methods.

## 4. How is the integral of Legendre polynomial related to Legendre-Gauss quadrature?

The integral of Legendre polynomial is used to derive the weights and nodes for Legendre-Gauss quadrature, which is a numerical method for approximating integrals. The Legendre-Gauss quadrature is based on the orthogonality property of Legendre polynomials and is highly accurate for integrating functions with smooth curves.

## 5. Are there any real-world examples that use the integral of Legendre polynomial?

Yes, the integral of Legendre polynomial has applications in various real-world scenarios, such as calculating the deflection of beams in mechanical engineering, determining the electric field in electrostatics, and analyzing the wave functions in quantum mechanics. It is also used in image processing, data compression, and signal processing.

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