/*! 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 16 Find the integral. $$ \int \... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 16

Find the integral. $$ \int \frac{x}{\sqrt{9-x^{2}}} d x $$

Short Answer

Expert verified
The integral of the given function, \( \int \frac{x}{\sqrt{9-x^2}} dx\) , is \(-\frac{1}{4} + \frac{x^2}{4} + C\).

Step by step solution

01

Initiate Substitution

We need to use trigonometric substitution. Let \(x = 3\sin \theta\). From the Pythagorean identity \(1 = \sin^2\theta + \cos^2\theta\), we have \(9-x^2 = 9\cos^2\theta\), then \(\sqrt{9-x^2} = 3\cos \theta\).
02

Calculate Derivative

Next, find the derivative of \(x = 3\sin \theta\) which is \(dx = 3\cos \theta d\theta\).
03

Replace dx in Integral

Substitute \(x = 3\sin \theta\) and \(dx = 3\cos \theta d\theta\) into the integral.
04

Simplify the Integral

The integral becomes \(\int \sin \theta \cdot 3\cos \theta d\theta\). With the double angle identity \(\sin 2\theta = 2\sin\theta \cos\theta\), it can be rewritten as \(\frac{1}{2}\int \sin 2\theta d\theta\).
05

Solve the Integral

The integral \(\frac{1}{2}\int \sin 2\theta d\theta\) simplifies to \(-\frac{1}{4}\cos 2\theta + C\).
06

Convert back to x Terms

Change the term back to \(x\) using the double angle identity \(\cos2\theta = 1- 2\sin^2\theta\). This will result in \(-\frac{1}{4} + \frac{x^2}{4} + C\).

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Most popular questions from this chapter

The Gamma Function \(\Gamma(n)\) is defined by \(\Gamma(n)=\int_{0}^{\infty} x^{n-1} e^{-x} d x, \quad n>0\) (a) Find \(\Gamma(1), \Gamma(2),\) and \(\Gamma(3)\). (b) Use integration by parts to show that \(\Gamma(n+1)=n \Gamma(n)\). (c) Write \(\Gamma(n)\) using factorial notation where \(n\) is a positive integer.

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