Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         
                                                                                                                                                                              
 ∫π/₁₆^π/⁸ 8 csc² 4𝓍 d𝓍

Verified step by step guidance

1. Step 1: Recognize that the integral involves the function csc²(4𝓍), which has a standard antiderivative. The antiderivative of csc²(u) is -cot(u).
2. Step 2: Perform a substitution to simplify the integral. Let u = 4𝓍, which implies that du = 4 d𝓍. Rewrite the integral in terms of u.
3. Step 3: Adjust the limits of integration according to the substitution. When 𝓍 = π/₁₆, u = 4(π/₁₆) = π/₄. When 𝓍 = π/₈, u = 4(π/₈) = π/₂.
4. Step 4: Rewrite the integral using the substitution. The integral becomes ∫π/₄^π/₂ 2 csc²(u) du, where the factor of 2 comes from dividing by 4 in the substitution.
5. Step 5: Evaluate the integral using the antiderivative of csc²(u). Substitute the limits of integration into -2 cot(u) and simplify.

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 to maintain the integrity of the integral.

Cosecant Function

The cosecant function, denoted as csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). In calculus, it often appears in integrals involving trigonometric functions. Understanding its properties and behavior is essential for evaluating integrals that include csc²(x), which is commonly encountered in integration problems.