Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

8. Definite Integrals

Substitution

Calculus

8. Definite Integrals

Substitution

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2:52 minutes

Problem 8.6.56

Textbook Question

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

Verified step by step guidance

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.

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.

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.

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.

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.

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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. 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.