In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(x - 2) / √(x - 1) dx

Verified step by step guidance

1. Start by examining the integral \(\int \frac{\sqrt{x - 2}}{\sqrt{x - 1}} \, dx\). Notice the expressions under the square roots are similar, differing by 1. This suggests a substitution to simplify the radicals.
2. Make the substitution \(u = \sqrt{x - 1}\). Then, express \(x\) in terms of \(u\): \(x = u^2 + 1\). Also, find \(dx\) in terms of \(du\): differentiate both sides to get \(dx = 2u \, du\).
3. Rewrite the integral in terms of \(u\). The numerator becomes \(\sqrt{x - 2} = \sqrt{u^2 + 1 - 2} = \sqrt{u^2 - 1}\), and the denominator is \(\sqrt{x - 1} = u\). Substitute these and \(dx\) into the integral to get \(\int \frac{\sqrt{u^2 - 1}}{u} \cdot 2u \, du\).
4. Simplify the integral expression: the \(u\) in the denominator and numerator cancel, leaving \(\int 2 \sqrt{u^2 - 1} \, du\). Now, to evaluate this integral, use a trigonometric substitution suitable for \(\sqrt{u^2 - 1}\), such as \(u = \sec \theta\).
5. With \(u = \sec \theta\), express \(\sqrt{u^2 - 1}\) as \(\sqrt{\sec^2 \theta - 1} = \tan \theta\), and find \(du = \sec \theta \tan \theta \, d\theta\). Substitute these into the integral and simplify to an integral in terms of \(\theta\) that can be evaluated using standard trigonometric integrals.

Key Concepts

Substitution Method in Integration

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It involves choosing a substitution that reduces complexity, often by setting a part of the integrand equal to a new variable, which helps in rewriting the integral in terms of this variable.

Trigonometric Substitution

Trigonometric substitution is used to evaluate integrals involving square roots of quadratic expressions. By substituting a trigonometric function for the variable, the integral is transformed using trigonometric identities, making it easier to integrate expressions like √(x - a) or √(a - x).

Integrals Involving Radical Expressions

Integrals with radicals often require special techniques because the square roots complicate direct integration. Recognizing the form of the radical and applying appropriate substitutions or trigonometric identities is essential to simplify and evaluate these integrals effectively.