Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         
                                                                                                                                                                              
 ∫₋₁¹ (𝓍―1) (𝓍²―2𝓍)⁷ d𝓍

Verified step by step guidance

1. Step 1: Recognize that the integral ∫₋₁¹ (𝓍 - 1)(𝓍² - 2𝓍)⁷ d𝓍 involves a composite function (𝓍² - 2𝓍) raised to a power. This suggests using a substitution method to simplify the integral.
2. Step 2: Let u = 𝓍² - 2𝓍. Compute the derivative of u with respect to 𝓍: du/d𝓍 = 2𝓍 - 2. Rewrite this as du = (2𝓍 - 2)d𝓍.
3. Step 3: Factorize the original integral to match the substitution. Notice that (𝓍 - 1) can be factored out from the derivative du = (2𝓍 - 2)d𝓍, since (𝓍 - 1) is a common factor. Rewrite the integral in terms of u.
4. Step 4: Change the limits of integration. When 𝓍 = -1, substitute into u = 𝓍² - 2𝓍 to find the lower limit of u. Similarly, when 𝓍 = 1, substitute into u = 𝓍² - 2𝓍 to find the upper limit of u.
5. Step 5: Rewrite the integral entirely in terms of u and evaluate using standard integration techniques or a table of integrals. After integration, substitute back the original variable if necessary.

Key Concepts

Definite Integrals

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The result of a definite integral is a number that quantifies the accumulation of quantities, such as area, volume, or total change, between the limits of integration.

Change of Variables

Change of variables, or substitution, is a technique used in integration to simplify the integrand 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 to maintain the integrity of the integral's value.

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 students to tackle a wide range of problems in calculus effectively.