Problem 37 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \( \sqrt{9\sec^2(\theta) - 9} = 3\tan(\theta) \).

Step 2: 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 \( x = 6 \), \( \theta = \sec^{-1}(6/3) = \sec^{-1}(2) \).

Step 3: 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 \( \sec^{-1}(2) \).

Step 4: 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.