please show integration steps i am stuck with the integration If $ f(x) $ is an even function, i.e., $ f(-x) = f(x) $, then its Fourier series reduces to the Fourier cosine series.\[f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos \left( \frac{n \pi}{L} x \right)\]\[ a_0 = \frac{1}{L} \int_0^L f(x) \, dx \]\[a_n = \frac{2}{L} \int_0^L f(x) \cos \left( \frac{n \pi}{L} x \right) dx \quad \text{for} \quad n = 1, 2, 3, \ldots\] **Problem 12.3.15:** Expand $ f(x) = x^2 $, $-1 < x < 1$ into an appropriate cosine or sine series.

Answered: Image /qna-images/answer/bb369a90-a514-4a19-8f5b-e2a060761d52.jpg

Answered: If f(x) is an even function, i.e. f(-x)…

If f(x) is an even function, i.e. f(-x) = f(x), then its Fourier series reduces to the Fourier cosine series. f(x )-ao + E an cos (Ex) E an COS 1 ao L f(x)dx 2 (п

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.

Question

please show integration steps i am stuck with the integration

If $ f(x) $ is an even function, i.e., $ f(-x) = f(x) $, then its Fourier series reduces to the Fourier cosine series.

\[
f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos \left( \frac{n \pi}{L} x \right)
\]

\[
a_0 = \frac{1}{L} \int_0^L f(x) \, dx
\]

\[
a_n = \frac{2}{L} \int_0^L f(x) \cos \left( \frac{n \pi}{L} x \right) dx \quad \text{for} \quad n = 1, 2, 3, \ldots
\]

**Problem 12.3.15:** Expand $ f(x) = x^2 $, $-1 < x < 1$ into an appropriate cosine or sine series.

With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.

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