7–84. Evaluate the following integrals.
56. ∫ from π to 3π/2 sin2x e^(sin²x) dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves a composition of functions, specifically sin²(x) and e^(sin²(x)). This suggests that substitution might be a useful technique to simplify the integral.
2. Step 2: Let u = sin²(x). Then, compute the derivative of u with respect to x: du/dx = 2sin(x)cos(x). This implies that du = 2sin(x)cos(x)dx.
3. Step 3: Rewrite the integral in terms of u. Notice that sin²(x) is replaced by u, and the differential dx is replaced by du/(2sin(x)cos(x)). The integral becomes ∫ e^u du/2.
4. Step 4: Adjust the limits of integration. When x = π, sin²(x) = sin²(π) = 0. When x = 3π/2, sin²(x) = sin²(3π/2) = 1. Thus, the new limits for u are from 0 to 1.
5. Step 5: Integrate ∫ e^u du/2 over the interval [0, 1]. This involves finding the antiderivative of e^u, which is e^u, and then evaluating it at the new limits of integration. Multiply the result by 1/2 to account for the factor outside the integral.

Key Concepts

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. In this case, the integral is evaluated from π to 3π/2, which means we are calculating the area under the curve of the function sin²x e^(sin²x) within that interval. Understanding how to set up and evaluate definite integrals is crucial for solving the problem.

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and recognizing patterns in integrals. For the given integral, recognizing that a substitution involving sin²x could simplify the expression is essential for finding the solution efficiently.

Exponential Functions

Exponential functions, such as e^(sin²x), are functions where the variable is in the exponent. They often arise in calculus problems involving growth and decay, and their properties can significantly affect the behavior of integrals. Understanding how to manipulate and integrate exponential functions is vital for evaluating the given integral.