Mathematical Integration and Series Lecture Notes

Key Concepts

1. Interval Definition

   • Interval: $[-\pi, \pi]$
   • Equal interval: 1

2. Even Functions

   • Integration properties and examples

Detailed Integrals and Calculations

Integral Example 1

• Given Function: $f(x) = x^2$
• Integral Calculation: [ \int_{0}^{\pi} x^2 dx = \frac{2}{\pi} \int_{0}^{\pi} x^2 dx = \frac{2}{\pi} \left[ \frac{x^3}{3} \right]_0^{\pi} ]
• Result: [ \frac{2}{\pi} \times \frac{\pi^3}{3} = \frac{2}{3} \pi^2 ]_

Integral Example 2

• Given Function: $f(x) = x^2 \cos(nx)$
• Integral Calculation:
  • Use integration by parts
  • Result includes summation notation and series analysis

Summation and Series

Summation of Series Example

• Series involved: $ \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$
  • Known result: $ \frac{\pi^2}{12}
  • Negative and positive term contributions

Another Series Analysis

• Example:
  • Various summations of series with $(-1)^n$ and $n^2$
  • Analysis of converging series and related values

Fourier Series Component Analysis

Sine and Cosine Functions

• Analysis of functions involving sine and cosine integrals
  • Examples with limits and integral results

Fourier Coefficients

• Calculation of Fourier coefficients: $B_n$
  • Example Formula: $B_n = \frac{4}{\pi n^2} \sin\left(\frac{n\pi}{2}\right)$

Conclusion

• Summarized integral results and values related to series
• Provided Fourier series examples and how periodic functions are addressed