Math Problem Statement

∫ cosh(x) * e^(-i * n * π * x) dx

Solution

The integral expression in the image appears to be:

cosh(x)einπxdx\int \cosh(x) e^{-i n \pi x} \, dx

To solve this, we need to focus on integrating the product of the hyperbolic cosine function cosh(x)\cosh(x) and the complex exponential function einπxe^{-i n \pi x}.

Step-by-Step Solution

  1. Recall the definition of cosh(x)\cosh(x):

cosh(x)=ex+ex2\cosh(x) = \frac{e^x + e^{-x}}{2}

  1. Substitute this into the integral:

cosh(x)einπxdx=(ex+ex2)einπxdx\int \cosh(x) e^{-i n \pi x} \, dx = \int \left(\frac{e^x + e^{-x}}{2}\right) e^{-i n \pi x} \, dx

  1. Distribute einπxe^{-i n \pi x} to both terms inside cosh(x)\cosh(x):

=12(exeinπx+exeinπx)dx= \frac{1}{2} \int \left(e^x e^{-i n \pi x} + e^{-x} e^{-i n \pi x}\right) \, dx =12(e(1inπ)x+e(1inπ)x)dx= \frac{1}{2} \int \left(e^{(1 - i n \pi)x} + e^{(-1 - i n \pi)x}\right) \, dx

  1. Integrate both exponential terms:

=12[e(1inπ)x1inπ+e(1inπ)x1inπ]+C= \frac{1}{2} \left[\frac{e^{(1 - i n \pi)x}}{1 - i n \pi} + \frac{e^{(-1 - i n \pi)x}}{-1 - i n \pi}\right] + C

Where CC is the constant of integration.

Thus, the final answer is:

12[e(1inπ)x1inπ+e(1inπ)x1inπ]+C\frac{1}{2} \left[\frac{e^{(1 - i n \pi)x}}{1 - i n \pi} + \frac{e^{(-1 - i n \pi)x}}{-1 - i n \pi}\right] + C

Would you like further clarification on any part of this solution?

Related Questions:

  1. How do we integrate exponential functions in general?
  2. What is the relationship between cosh(x)\cosh(x) and exe^x?
  3. Can we extend this solution to integrals involving other hyperbolic functions?
  4. What happens if n=0n = 0 in this integral?
  5. How does the complex exponential function eiθe^{i\theta} relate to trigonometric functions?

Tip:

When integrating products of exponentials, it's useful to remember that exponentials with linear terms in the exponent can be integrated directly by dividing by the linear term.

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Math Problem Analysis

Mathematical Concepts

Hyperbolic functions
Exponential functions
Complex numbers

Formulas

cosh(x) = (e^x + e^(-x)) / 2
Integral of e^(ax) dx = e^(ax) / a

Theorems

Fundamental Theorem of Calculus
Linear combination of exponentials

Suitable Grade Level

Undergraduate (Calculus/Complex Analysis)