Chapter 8: Problem 14

Define the integrals \(I_{n}=\int_{-\infty}^{\infty} x^{2 n} e^{-x^{2}} d x .\) Noting that \(I_{0}=\sqrt{\pi}\) a. Find a recursive relation between \(I_{n}\) and \(I_{n-1} .\) b. Use this relation to determine \(I_{1}, I_{2}\), and \(I_{3}\). c. Find an expression in terms of \(n\) for \(I_{n}\).

Short Answer

Expert verified

The recursive relation is \(I_{n} = 2nI_{n-1}\) using this result to the given \(I_{0} = \sqrt{\pi}\) we find \(I_{1} = \sqrt{\pi}\), \(I_{2} = 2\sqrt{\pi}\) and \(I_{3} = 12\sqrt{\pi}\). Hence, a generalized formula can be derived: \(I_{n} = n!(2^{n})\sqrt{\pi}\)

Step by step solution

Define The Recursive Relation

To find the recursive relation, integrating by parts is required. Using the formula \(\int u dv = uv - \int v du\), let \(u = x^{2n}\) and \(dv = e^{-x^{2}} dx\). Calculate \(du\) and \(v\), and substitute these values into the integration by parts formula. This will help to establish the relationship between \(I_{n}\) and \(I_{n-1}\).

Use The Recursive Relation

Once the recursive relation from step one has been found, it should be used to determine the values of \(I_{1}, I_{2}\), and \(I_{3}\). To do this, start with the provided value of \(I_{0}\) which is equal to \(\sqrt{\pi}\) and then apply the recursive relation iteratively for \(n = 1, 2, 3\).

Find a General Expression

In this step, upon evaluation of the values of \(I_{1}, I_{2}\) and \(I_{3}\), observe the pattern and formulate a general term \(I_{n}\) based on the pattern observed. This will involve using the concept of mathematical induction and pattern recognition.