7–64. Integration review Evaluate the following integrals.
40. ∫ (1 - x) / (1 - √x) dx

Verified step by step guidance

1. Rewrite the integrand to simplify the expression. Start by factoring or rationalizing the denominator if necessary. For this problem, consider rewriting the square root term \( \sqrt{x} \) as \( u \), where \( u = \sqrt{x} \) and \( x = u^2 \).
2. Perform a substitution: Let \( u = \sqrt{x} \), so \( dx = 2u \, du \). Substitute \( x = u^2 \) and \( dx \) into the integral, transforming it into terms of \( u \).
3. Simplify the new integral in terms of \( u \). Replace \( 1 - x \) with \( 1 - u^2 \) and \( 1 - \sqrt{x} \) with \( 1 - u \). The integral becomes \( \int \frac{1 - u^2}{1 - u} \cdot 2u \, du \).
4. Simplify the fraction \( \frac{1 - u^2}{1 - u} \) by factoring \( 1 - u^2 \) as \( (1 - u)(1 + u) \). Cancel the \( 1 - u \) terms, leaving \( 1 + u \). The integral now becomes \( \int 2u(1 + u) \, du \).
5. Expand the integrand \( 2u(1 + u) \) to \( 2u + 2u^2 \), and integrate term by term. Use the power rule for integration: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). After integrating, substitute back \( u = \sqrt{x} \) to return to the original variable.

Key Concepts

Integration Techniques

Integration techniques are methods used to evaluate integrals, which can include substitution, integration by parts, and partial fraction decomposition. Understanding these techniques is essential for simplifying complex integrals into manageable forms. In this case, recognizing the structure of the integrand can guide the choice of the appropriate technique.

Substitution Method

The substitution method is a powerful technique in integration that involves changing the variable of integration to simplify the integral. By substituting a part of the integrand with a new variable, the integral can often be transformed into a more straightforward form. For the given integral, a suitable substitution can help eliminate the square root in the denominator.

Rational Functions

Rational functions are ratios of polynomials, and their integration often requires specific strategies, such as polynomial long division or partial fraction decomposition. In the integral provided, recognizing that the integrand is a rational function allows for the application of these strategies to break it down into simpler components that can be integrated individually.