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

7. Antiderivatives & Indefinite Integrals

Integrals of Trig Functions

Calculus

7. Antiderivatives & Indefinite Integrals

Integrals of Trig Functions

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2:31 minutes

Problem 8.1.69

Textbook Question

69. Different substitutions
b. Evaluate ∫(tan x sec² x) dx using the substitution u=secx.

Verified step by step guidance

Step 1: Identify the substitution given in the problem. Here, we are instructed to use u = sec(x). This substitution will help simplify the integral.

Step 2: Compute the derivative of u with respect to x. Since u = sec(x), we have du/dx = sec(x)tan(x). Therefore, du = sec(x)tan(x) dx.

Step 3: Rewrite the integral ∫(tan(x) sec²(x)) dx using the substitution u = sec(x). Replace sec(x) with u and tan(x) sec(x) dx with du.

Step 4: After substitution, the integral becomes ∫u du because sec²(x) becomes u² and tan(x) sec(x) dx is replaced by du.

Step 5: Integrate ∫u du to find the antiderivative. The result will be expressed in terms of u, and then substitute back u = sec(x) to express the final answer in terms of x.

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Key Concepts

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

Integration

Integration is a fundamental concept in calculus that involves finding the antiderivative of a function. It is the process of calculating the area under a curve represented by a function over a specified interval. Understanding integration is crucial for solving problems involving accumulation and area, and it often requires techniques such as substitution to simplify the integrand.

Substitution Method

The substitution method is a technique used in integration to simplify complex integrals. By substituting a part of the integrand with a new variable, the integral can often be transformed into a more manageable form. In this case, using the substitution u = sec(x) allows us to express the integral in terms of u, making it easier to evaluate.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are essential for simplifying expressions and solving integrals involving trigonometric functions. In this problem, recognizing the relationship between tan(x), sec(x), and their derivatives is key to applying the substitution effectively.

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