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

∫₀^π/⁴ eˢᶦⁿ² ˣ sin 2𝓍 d𝓍

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Step 1: Recognize that the integral involves a composition of functions, specifically e^(sin²(x)) and sin(2x). To simplify, consider using a substitution method to reduce the complexity of the integral.

Step 2: Let u = sin(x). Then, du = cos(x) dx. This substitution will help simplify the integral. Also, update the limits of integration: when x = 0, u = sin(0) = 0; when x = π/4, u = sin(π/4) = √2/2.

Step 3: Rewrite sin²(x) in terms of u. Since u = sin(x), sin²(x) becomes u². Additionally, sin(2x) can be expressed as 2sin(x)cos(x), which simplifies to 2u√(1-u²) using the substitution.

Step 4: Substitute these expressions into the integral. The integral becomes ∫₀^(√2/2) e^(u²) * 2u√(1-u²) du. This is now in terms of u, and the limits of integration are updated accordingly.

Step 5: Evaluate the integral using either a table of integrals (Table 5.6) or numerical methods, as the integral involves a non-trivial composition of functions. The presence of e^(u²) and √(1-u²) suggests that further simplification or approximation may be required.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 definite integrals is crucial for evaluating areas, volumes, and other physical quantities.

Change of Variables

The change of variables technique, also known as substitution, is a method 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 technique is particularly useful when dealing with complex functions or when the integral involves compositions of functions, allowing for easier integration and evaluation.

Integration Techniques

Integration techniques encompass various methods used to compute integrals, including substitution, integration by parts, and using integral tables. These techniques are essential for solving integrals that cannot be evaluated using basic antiderivatives. Familiarity with these methods, such as those found in Table 5.6, allows students to efficiently tackle a wide range of integral problems, including those involving trigonometric, exponential, and logarithmic functions.

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