Problem 30 - Calculus: One and Several Variables

Question

Find a power series representation for the improper integral. $$\int_{0}^{x} \frac{\sinh t}{t} d t$$

Step-by-Step Solution

Answer

The power series representation for the improper integral \(\int_{0}^{x} \frac{\sinh t}{t} d t\) is given by \(\sum_{n=0}^{+\infty} \frac{{x^{2n+1}}}{(2n+1)! * (2n+1)}\)

Step 1: Express \(\sinh(t)\) as a power series

The hyperbolic sine function, \(\sinh(t)\), can be expanded into a power series. We write this as follows: \(\sinh(t) = t + \frac{{t^3}}{3!} + \frac{{t^5}}{5!} + \cdots = \sum_{n=0}^{\infty} \frac{{t^{2n+1}}}{(2n+1)!}\)

Step 2: Substitute power series representation for \(\sinh(t)\) into integral

Substituting this expansion of \(\sinh(t)\) into the given integral simplifies it to: \(\int_{0}^{x} \frac{\sinh t}{t} d t = \int_0^x \left(\sum_{n=0}^{+\infty} \frac{{t^{2n}}}{(2n+1)!}\right) dt\)

Step 3: Integrate term-by-term

The next step is to integrate this power series term-by-term. This gives: \(\int_0^x \left(\sum_{n=0}^{+\infty} \frac{{t^{2n}}}{(2n+1)!}\right) dt = \sum_{n=0}^{+\infty} \left(\int_0^x \frac{{t^{2n}}}{(2n+1)!} dt\right) = \sum_{n=0}^{+\infty} \frac{{x^{2n+1}}}{(2n+1)! * (2n+1)}\)

Key Concepts

Improper Integral

An improper integral is a type of integral where either the function being integrated or the limits of integration are unbounded. In simpler terms, we’re dealing with an infinite process, either because we’re integrating over an infinite interval, or because the function has a discontinuity or goes to infinity within the integration limits.
In this case, \[\int_{0}^{x} \frac{\sinh t}{t} d t\] is considered improper because it involves dividing by zero when \(t = 0\), which is not defined in standard calculus. However, improper integrals can often still be computed by finding a suitable way to handle the 'problematic' parts of the integral — like using a power series representation to break it down into simpler parts.

Hyperbolic Functions

Just like the familiar trigonometric functions, hyperbolic functions are an important class of mathematical functions. They are used to describe the shapes of hyperbolas, much like trigonometric functions are related to circles.
The hyperbolic sine function, denoted as \(\sinh(t)\), is defined as \(\sinh(t) = \frac{e^t - e^{-t}}{2}\). It is often used in the study of hanging cables (called catenary), flight paths, and even special relativity. Our integral uses \(\sinh(t)\) and by expressing it as a power series — \(\sinh(t) = t + \frac{{t^3}}{3!} + \frac{{t^5}}{5!} + \cdots\) — we can integrate it more easily, especially term-by-term.

Term-by-Term Integration

The method of term-by-term integration allows us to integrate power series by integrating each term individually. This is a powerful technique because it transforms the challenging task of integrating a complex function into the sum of many simpler integrals.
Applying this method to our problem, we integrate each term of the power series representing \(\sinh(t)\). Here, it is crucial that the power series converges on the interval of integration, which in this case, it does. We end up with a series of terms that are more straightforward to integrate because they are just powers of \(t\) divided by a constant.

Infinite Series

In mathematics, an infinite series is the sum of infinitely many terms, and it can be denoted as \(\sum_{n=0}^{\infty} a_n\), where \(a_n\) represents the n-th term of the series. These series can converge to a finite value or diverge to infinity, depending on their terms.
The power series representation of \(\sinh(t)\) is an example of an infinite series that can be used to describe functions which may not have simple antiderivatives. In the context of our integral, the power series helps us to find a representation for the integral as another infinite series, allowing us to see the behavior of the integral as \(x\) varies — invaluable for understanding complex functions over a domain.