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Chapter 6: Problem 37

Evaluate the integral using (a) the given integration limits and (b) the limits obtained by trigonometric substitution. $$ \int_{4}^{6} \frac{x^{2}}{\sqrt{x^{2}-9}} d x $$

Short Answer

Expert verified
The step-by-step solution presented clearly exemplifies how to solve the given integral problem by explicitly using trigonometric substitution.

Step by step solution

01

Identify The Trigonometric Substitution

In this case, where the integral has the form \( \int \frac{x^2}{\sqrt{x^{2}-a^{2}}} dx \), where \( a = 3 \), a suitable substitution is \( x = 3\sec(\theta) \). Considering \( dx = 3\sec(\theta)\tan(\theta)d\theta \) and \( x^2 = 9\sec^2(\theta) \). The square root in the denominator now becomes \( \sqrt{9\sec^2(\theta) - 9} = 3\tan(\theta) \).
02

Replace in The Original Integral

Substitute x with the identified trigonometric function and dx with its derivative: \( \int \frac{9\sec^2(\theta)}{3\tan(\theta)} 3\sec(\theta)\tan(\theta)d\theta\). Which further simplifies to \( \int \frac {9\sec^2(\theta)}{\tan(\theta)}\sec(\theta)\tan(\theta) d\theta = 9\int \sec^3(\theta)d\theta \). The limits of integration now change to match \( \theta \), since we are integrating with respect to \( \theta \). When \( x = 4 \), \( \theta = \sec^{-1}(4/3)\) and when \( x = 6 \), \( \theta = \sec^{-1}(6/3) = \sec^{-1}(2) \).
03

Solve The Integral

The integral \( \int \sec^3(\theta)d\theta \) simplifies to \( \frac{1}{2}\sec(\theta)\tan(\theta) + \frac{1}{2}\ln| \sec(\theta) + \tan(\theta)|\). We integrate this from \( \sec^{-1}(4/3) \) to \( \sec^{-1}(2) \).
04

Substitution of The Trigonometric Limit and Simplification

Lastly, substitute the limits of \( \theta \), back into the integral and simplify the resulting expression. This should yield the final result.

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Most popular questions from this chapter

(a) find the indefinite integral in two different ways. (b) Use a graphing utility to graph the antiderivative (without the constant of integration) obtained by each method to show that the results differ only by a constant. (c) Verify analytically that the results differ only by a constant. $$ \int \sec ^{2} x \tan x d x $$

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