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Chapter 8: Problem 48

Use an appropriate substitution and then a trigonometric substitution to evaluate the integrals. $$\int \frac{\sqrt{x-2}}{\sqrt{x-1}} d x$$

Short Answer

Expert verified
The integral evaluates to \( 2[\sqrt{x-2} - \arctan(\sqrt{x-2})] + C \).

Step by step solution

01

Identify the Substitution

The integral given is \( \int \frac{\sqrt{x-2}}{\sqrt{x-1}} \, dx \). We notice that both the numerator and the denominator involve square roots of linear transformations of \( x \). To simplify the integrand, consider the substitution \( x = u + 2 \). This will eliminate the subtraction under the square root. As a result, the transformation \( x - 2 = u \) and \( dx = du \) can be applied.
02

Change Variables Using the Substitution

Using the substitution \( x = u+2 \), the integral becomes \( \int \frac{\sqrt{u}}{\sqrt{u+1}} \, du \). Now we need to simplify the integrand further. For trigonometric substitution, it is helpful to express this in terms of a trigonometric identity. Hence, consider expressing \( u+1 \) in terms of a trigonometric function.
03

Apply a Trigonometric Substitution

We use the trigonometric substitution \( u = \tan^2(\theta) \), which implies \( du = 2\tan(\theta)\sec^2(\theta) \, d\theta \). The integrand \( \int \frac{\sqrt{\tan^2(\theta)}}{\sqrt{\tan^2(\theta) + 1}}\cdot 2\tan(\theta)\sec^2(\theta) \, d\theta \) becomes \( \int 2\tan^2(\theta) \, d\theta \), because \( \sqrt{\tan^2(\theta)}=\tan(\theta) \) and \( \sqrt{\tan^2(\theta) + 1}=\sec(\theta) \).
04

Simplify and Integrate

The integral \( \int 2\tan^2(\theta) \, d\theta \) can be simplified using the identity \( \tan^2(\theta) = \sec^2(\theta) - 1 \). Thus the integral becomes \( \int 2(\sec^2(\theta) - 1) \, d\theta = 2\int (\sec^2(\theta) - 1) \, d\theta \). Integrating \( \int \sec^2(\theta) \, d\theta \) gives \( \tan(\theta) \), and \( \int d\theta \) gives \( \theta \). Therefore, \( 2[\tan(\theta) - \theta] + C \).
05

Reverse the Substitutions

After integration, we revert the trigonometric substitution. Recall that \( u = \tan^2(\theta) \) implies \( \theta = \arctan(\sqrt{u}) \), and \( u = x - 2 \). Substitute back to get \( 2[\sqrt{x-2} - \arctan(\sqrt{x-2})] + C \). This represents the final expression of the integral evaluated.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration techniques
When dealing with complex integrals, like \( \int \frac{\sqrt{x-2}}{\sqrt{x-1}} dx \), it is essential to familiarize yourself with various integration techniques to simplify them. Key techniques include:
  • Substitution: Useful for reducing the complexity of the integrand by changing variables.
  • Trigonometric Substitution: Allows us to handle integrals involving square roots, often converting them into forms that are easier to integrate.
  • Recognizing and Applying Identities: Helps in simplifying the expressions within the integral.
The idea is to transform the original integral into a more manageable form so that your integration becomes straightforward. By combining these techniques systematically, you can tackle otherwise challenging problems with confidence.
Substitution method
The substitution method simplifies integrals by converting them into a form that can be more easily integrated. Let's consider the given exercise: \( \int \frac{\sqrt{x-2}}{\sqrt{x-1}} dx \). Here鈥檚 how we tackle it:
  • Identify the Inner Function: We notice \( x-2 \) and \( x-1 \) are linear transformations. So, we use \( x = u + 2 \). Thus, \( x - 2 = u \) and \( dx = du \). The integral transforms to \( \int \frac{\sqrt{u}}{\sqrt{u+1}} du \).
  • New Integrand Formation: This 'u' substitution has now simplified the variables inside the radical expressions and prepared it for a trigonometric substitution.
Proper substitution effectively reduces complexity, setting the stage for further simplification through additional methods.
Trigonometric identities
Trigonometric identities are powerful tools when working with trigonometric substitution in integration. They help to further simplify the integrals, making them easier to calculate.In our task, once we applied \( u = \tan^2(\theta) \), we transformed the complex radical expressions into trigonometric functions. This is possible because of key trigonometric identities:
  • Basic Identity: \( \tan^2(\theta) + 1 = \sec^2(\theta) \) allows us to replace and simplify expressions under the square roots.
  • Simplification: In terms of \( \theta \), the integral \( \int 2\tan^2(\theta) d\theta \) becomes manageable because we can use \( \tan^2(\theta) = \sec^2(\theta) - 1 \).
  • Direct Integration: Integrate \( \sec^2(\theta) \) directly to get \( \tan(\theta) \), transforming the expression significantly.
By incorporating these identities, you can execute the integration seamlessly, ultimately transitioning back to the original variable with ease.

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