/*! 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 18 Find the integral. $$ \int x... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 18

Find the integral. $$ \int x \sqrt{16-4 x^{2}} d x $$

Short Answer

Expert verified
The integral of \(x \sqrt{16-4x^{2}}\) dx is \(-8x \sqrt{16-x^2} + \frac{8}{3}(16-x^2)^{\frac{3}{2}} + C\) where C is the constant of integration.

Step by step solution

01

Identify the Correct Substitution

The denominator part of the integrand under the square root \(\sqrt{16-4x^2}\) resembles the formula for Pythagorean identity in trigonometry \(\sin^2\theta + \cos^2\theta = 1\). For easy handling of the integral, one introduces a substitution \(2x=4\sin\theta\) or \(x=2\sin\theta\). The differential dx can be computed as \(dx=2\cos\theta d\theta\).
02

Substitute and Simplify the Integral

Let's substitute \(x=2\sin\theta\) and \(dx=2\cos\theta d\theta\) into the integral \[ \int x \sqrt{16-4x^2} dx = \int 2\sin\theta\sqrt{16-16\sin^2\theta} * 2\cos\theta d\theta.\] After simplifying, we get \[ \int 2\sin\theta*(4|\cos\theta|)*2\cos\theta d\theta = 8\int\sin\theta \cos^2\theta d\theta.\] Here, \( |\cos\theta|\) is taken as \(\cos\theta\) since the restriction that we have on \(\theta\), after substitution is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) which means \(\cos \theta >= 0 \) avoiding the need to split the integral.
03

Use Trigonometric Identity

Next, we can express \(\cos^2\theta\) in terms of \(\sin\theta\) using the Pythagorean identity \(\cos^2\theta = 1 - \sin^2\theta\), and the integral becomes\[ \int 8\sin\theta (1-\sin^2\theta) d\theta = 8 \int (\sin\theta-\sin^3\theta) d\theta.\]
04

Evaluate the Integral

The integral can now be integrated directly, yielding\[ 8 \int (\sin\theta - \sin^3\theta) d\theta = 8 (-cos\theta + \frac{cos^3\theta}{3}) + C, \]where \(C\) is the constant of integration.
05

Back-Substitute to Original Variable

Now we should express our answer in terms of the original variable \(x\). Remembering that we used the substitution \(x=2\sin\theta\), to find \(\cos\theta\), we use \(\cos\theta = \sqrt{1-\sin^2\theta} = \sqrt{1-\frac{x^2}{4}}\), we substitute back and simplify, finally we reach that the integral is equal to\[ -8x \sqrt{16-x^2} + \frac{8}{3}(16-x^2)^{\frac{3}{2}} + C.\]

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