In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ (√2 - x) / √x dx

Verified step by step guidance

1. Identify a substitution that simplifies the integral. Since the integral contains both \( \sqrt{x} \) and \( x \), let’s set \( u = \sqrt{x} \). This means \( u = x^{1/2} \).
2. Express \( x \) and \( dx \) in terms of \( u \). Since \( u = x^{1/2} \), then \( x = u^2 \). Differentiate both sides with respect to \( u \) to find \( dx \): \( dx = 2u \, du \).
3. Rewrite the integral in terms of \( u \). Substitute \( \sqrt{x} = u \), \( x = u^2 \), and \( dx = 2u \, du \) into the integral: \[ \int \frac{\sqrt{2} - x}{\sqrt{x}} \, dx = \int \frac{\sqrt{2} - u^2}{u} \cdot 2u \, du \].
4. Simplify the integrand by canceling and combining terms: \( \frac{\sqrt{2} - u^2}{u} \cdot 2u = 2(\sqrt{2} - u^2) \). So the integral becomes \( \int 2(\sqrt{2} - u^2) \, du \).
5. Now, split the integral into two simpler integrals: \[ 2 \int \sqrt{2} \, du - 2 \int u^2 \, du \]. These are standard integrals that can be found in the integral table and evaluated easily.

Key Concepts

Substitution Method in Integration

The substitution method involves changing variables in an integral to simplify it into a more familiar form. By letting a new variable equal a function of the original variable, the integral can be rewritten and evaluated more easily, often matching standard integral forms.

Manipulating Radicals and Algebraic Expressions

Understanding how to rewrite expressions involving square roots and algebraic terms is essential. This includes expressing radicals as fractional exponents and simplifying the integrand to facilitate substitution and integration.

Using Integral Tables

Integral tables provide formulas for common integrals, allowing quick evaluation once the integral is transformed appropriately. Recognizing the form of the integral after substitution helps in matching it to a table entry for direct integration.