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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2:43 minutes

Problem 8.3.40

Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.
40. ∫[0 to π/6] tan⁵(2x) sec(2x) dx

Verified step by step guidance

Step 1: Recognize that the integral involves a combination of trigonometric functions, specifically tan⁵(2x) and sec(2x). To simplify, recall that the derivative of tan(u) is sec²(u), which suggests a substitution method might be effective.

Step 2: Let u = tan(2x). Then, the derivative of u with respect to x is du/dx = 2 sec²(2x). Rearrange this to express dx in terms of du: dx = du / (2 sec²(2x)).

Step 3: Rewrite the integral using the substitution u = tan(2x). Since tan⁵(2x) = u⁵ and sec(2x) remains in the integral, substitute dx as well: ∫[0 to π/6] tan⁵(2x) sec(2x) dx = ∫[0 to π/6] u⁵ sec(2x) * (du / (2 sec²(2x))).

Step 4: Simplify the integral by canceling one sec(2x) term in the numerator with one in the denominator, leaving: ∫ u⁵ * (1 / 2 sec(2x)) du. Note that sec²(2x) is related to tan²(2x) via the identity sec²(θ) = 1 + tan²(θ), which can be used to express sec²(2x) in terms of u.

Step 5: Adjust the limits of integration to match the substitution. When x = 0, u = tan(2 * 0) = 0. When x = π/6, u = tan(2 * π/6) = tan(π/3). The integral now becomes ∫[0 to tan(π/3)] u⁵ / 2 du, which can be solved by applying the power rule for integration.

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Key Concepts

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

Trigonometric Functions

Trigonometric functions, such as tangent (tan) and secant (sec), are fundamental in calculus, particularly in integration. The tangent function is defined as the ratio of the sine and cosine functions, while the secant function is the reciprocal of the cosine function. Understanding their properties and relationships is crucial for evaluating integrals involving these functions.

Integration Techniques

Integration techniques, such as substitution and integration by parts, are essential for solving complex integrals. In this case, recognizing that the derivative of sec(2x) is tan(2x) sec(2x) allows for a substitution that simplifies the integral. Mastery of these techniques enables students to tackle a variety of integral forms effectively.

Definite Integrals

Definite integrals calculate the area under a curve between two specified limits, in this case, from 0 to π/6. The evaluation of definite integrals involves finding the antiderivative of the integrand and then applying the Fundamental Theorem of Calculus. Understanding how to compute definite integrals is crucial for solving problems in calculus, especially those involving trigonometric functions.