/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 465 Express \(\mathrm{x} \int_{0} \s... [FREE SOLUTION] | 91影视

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Chapter 19: Problem 465

Express \(\mathrm{x} \int_{0} \sqrt{\left(1-4 \sin ^{2} \mathrm{u}\right) \text { du in terms of incomplete elliptic }}\) integrals where \(0 \leq \mathrm{x} \leq(\pi / 6)\).

Short Answer

Expert verified
The given integral can be expressed in terms of an incomplete elliptic integral of the second kind as \[\mathrm{x} E(\mathrm{u}, 2), \quad 0 \leq \mathrm{u} \leq (\pi / 6).\]

Step by step solution

01

Recall the definition of an incomplete elliptic integral of the second kind

An incomplete elliptic integral of the second kind, denoted as \(E(\phi, k)\), is defined as: \[E(\phi , k) = \int_{0}^{\phi} \sqrt{1 - k^2 \sin^2 \theta} \mathrm{ d \theta}.\]
02

Compare the given integral to the elliptic integral definition

Given integral is: \[\mathrm{x} \int_{0} \sqrt{\left(1-4 \sin ^{2} \mathrm{u}\right)} \mathrm{du}.\] Comparing it with the definition of an elliptic integral of the second kind: \[E(\phi , k) = \int_{0}^{\phi} \sqrt{1 - k^2 \sin^2 \theta} \mathrm{ d \theta},\] We can notice the similarities as both the integrals have the same integral form. Now, in order to rewrite the given integral in terms of elliptic integrals, we need to identify \(\phi\) and \(k\).
03

Identify \(\phi\) and \(k\) from the given integral

The given integral is: \[\mathrm{x} \int_{0} \sqrt{\left(1-4 \sin ^{2} \mathrm{u}\right)} \mathrm{du}.\] Comparing it with the definition of an elliptic integral of the second kind \[E(\phi , k) = \int_{0}^{\phi} \sqrt{1 - k^2 \sin^2 \theta} \mathrm{ d \theta},\] we can make the following identifications: 1. \(\phi = \mathrm{u}\) 2. \(k^2 = 4\) With these identifications, we have \(k = 2\).
04

Rewrite the given integral in terms of elliptic integrals

Now that we have identified \(\phi\) and \(k\), we can rewrite the given integral in terms of the incomplete elliptic integral of the second kind. The given integral is: \[\mathrm{x} \int_{0} \sqrt{\left(1-4 \sin ^{2} \mathrm{u}\right)} \mathrm{du}.\] Using our identifications from Step 3, we can express the integral as: \[\mathrm{x} E(\mathrm{u}, 2).\] Since \(0 \leq \mathrm{x} \leq (\pi / 6)\), we can conclude: \[0 \leq \mathrm{u} \leq (\pi / 6).\] So, the expression of the given integral in terms of an incomplete elliptic integral of the second kind is: \[\mathrm{x} E(\mathrm{u}, 2), \quad 0 \leq \mathrm{u} \leq (\pi / 6).\]

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