71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.
71. ∫[x/(ax + b)] dx (Hint: u = ax + b.)

Verified step by step guidance

1. Step 1: Recognize that the integral ∫[x/(ax + b)] dx can be simplified using substitution. Let u = ax + b, which implies that du = a dx.
2. Step 2: Rewrite x in terms of u. Since u = ax + b, solving for x gives x = (u - b)/a.
3. Step 3: Substitute u and du into the integral. Replace x with (u - b)/a and dx with du/a, transforming the integral into ∫[((u - b)/a) / u] * (1/a) du.
4. Step 4: Simplify the integral. Factor out constants (1/a²) and split the fraction into two terms: ∫[(u/a²) / u] du - ∫[(b/a²) / u] du.
5. Step 5: Evaluate each term separately. The first term simplifies to ∫(1/a²) du, and the second term simplifies to ∫(-b/(a²u)) du. Use the standard integral formulas for these terms to proceed further.

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function. It is the reverse process of differentiation and can be used to calculate quantities such as total distance, area, and volume. Understanding integration techniques, such as substitution, is essential for solving integral problems.

Substitution Method

The substitution method is a technique used in integration to simplify the process of finding an integral. By substituting a part of the integrand with a new variable (often denoted as 'u'), the integral can be transformed into a more manageable form. In the given problem, substituting 'u = ax + b' simplifies the integral, making it easier to evaluate.

Rational Functions

Rational functions are ratios of polynomials, and they often appear in integration problems. The integral of a rational function can sometimes be simplified using algebraic manipulation or substitution. In this case, the function x/(ax + b) is a rational function, and understanding its structure is crucial for applying the appropriate integration techniques.