Calculate the definite integral of x/(cosh(x))^n from 0 to infinity.

Published on December 5, 2024

Calculus

The integral \int_{0}^{\infty} \frac{x}{(\cosh x)^2} dx evaluates to ln(2).

Question

Integration from zero to infinity of x/(coshx)^n

Answer

Here’s how to solve the integral:

1. Understanding the Problem

We want to evaluate

2. Rewriting cosh(x)

Recall the definition of the hyperbolic cosine:

So,

3. Expanding the denominator for the case n=2

Let’s consider the case when

for simplicity.

4. Substitution

Let

. Then

, so

. Also,

. When

. The integral becomes:

5. Integration by parts

Let

. Let

, so

The first term evaluates to 0 as

by L’Hopital’s rule.

6. Partial Fractions

Therefore, for

, the integral evaluates to

7. General Case

For general

, the integral is difficult to evaluate in closed form. It can be expressed in terms of special functions, but a simple elementary solution isn’t available. The integral can be evaluated numerically for specific values of

Final Answer: The answer is

when

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