Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.

∫π/₄^π/² (cos 𝓍) / (sin² 𝓍) d𝓍

검증된 단계별 안내

Step 1: Recognize that the integral involves a trigonometric function ratio. Rewrite the integrand (cos(𝓍) / sin²(𝓍)) in terms of simpler trigonometric expressions. Notice that cos(𝓍) / sin²(𝓍) can be expressed as (1 / sin²(𝓍)) * cos(𝓍).

Step 2: Use substitution to simplify the integral. Let u = sin(𝓍), which implies that du = cos(𝓍) d𝓍. This substitution transforms the integral into ∫ (1 / u²) du.

Step 3: Rewrite the limits of integration in terms of u. When 𝓍 = π/₄, u = sin(π/₄) = √2/2. When 𝓍 = π/₂, u = sin(π/₂) = 1. The new limits of integration are from u = √2/2 to u = 1.

Step 4: Integrate the transformed function ∫ (1 / u²) du. Recall that the integral of 1/u² is -1/u. Apply this formula to compute the antiderivative.

Step 5: Evaluate the definite integral by substituting the limits of integration into the antiderivative. Compute the result as [-1/u] evaluated from u = √2/2 to u = 1.

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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 number that quantifies the accumulation of the function's values over the interval [a, b]. Understanding how to evaluate definite integrals is crucial for solving problems in calculus.

Change of Variables

Change of variables, or substitution, is a technique used to simplify the evaluation of integrals. By substituting a new variable for an existing one, the integral can often be transformed into a more manageable form. This method involves calculating the derivative of the new variable and adjusting the limits of integration accordingly. It is particularly useful when dealing with complex functions or integrals that are difficult to evaluate directly.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. These identities, such as sin²(x) + cos²(x) = 1, can be used to simplify integrals involving trigonometric functions. Recognizing and applying these identities can make it easier to manipulate and evaluate integrals, especially when they appear in definite integrals like the one presented in the question.

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. ∫π/₄^π/² (cos 𝓍) / (sin² 𝓍) d𝓍