Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         
                                                                                                                                                                              
 ∫₁/₃^¹/√³ 4/(9𝓍² + 1) d𝓍

Verified step by step guidance

1. Step 1: Recognize that the integral ∫₁/₃^¹/√³ (4 / (9𝓍² + 1)) d𝓍 can be evaluated using a substitution method or by referencing a standard integral formula from a table. The denominator resembles the form of a standard integral involving arctangent.
2. Step 2: Perform a substitution to simplify the integral. Let u = 3𝓍, which implies du = 3 d𝓍. Rewrite the integral in terms of u: the limits of integration change accordingly. When 𝓍 = 1/3, u = 1, and when 𝓍 = 1/√3, u = √3.
3. Step 3: Substitute into the integral. The integral becomes ∫₁^√³ (4 / (u² + 1)) (1/3) du, where the factor 1/3 comes from the substitution du = 3 d𝓍.
4. Step 4: Factor out constants from the integral. The integral simplifies to (4/3) ∫₁^√³ (1 / (u² + 1)) du. This integral matches the standard formula for the arctangent function: ∫ (1 / (u² + 1)) du = arctan(u).
5. Step 5: Apply the formula for the arctangent function. Evaluate (4/3) [arctan(u)] from u = 1 to u = √3. Substitute the limits into the arctangent function to complete the evaluation.

Key Concepts

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. It is denoted as ∫[a, b] f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The result of a definite integral is a numerical value that quantifies the accumulation of the function's values over the interval [a, b].

Change of Variables

Change of variables, or substitution, is a technique used in integration to simplify the integral by transforming it into a more manageable form. This involves substituting a new variable for an existing one, which can make the integral easier to evaluate. The process requires adjusting the limits of integration and the differential accordingly.

Integration Techniques

Integration techniques encompass various methods used to evaluate integrals, including substitution, integration by parts, and using tables of integrals. These techniques are essential for solving complex integrals that cannot be evaluated using basic antiderivatives. Familiarity with these methods allows for more efficient and effective integration.