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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2:10 minutes

Problem 8.3.45

Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.
45. ∫ sec²x tan¹ᐟ²x dx

Verified step by step guidance

Step 1: Recognize the integral involves trigonometric functions sec²x and tan¹ᐟ²x. Recall that sec²x is the derivative of tan(x), which suggests a substitution method might be useful.

Step 2: Let u = tan(x). Then, du = sec²x dx. This substitution simplifies the integral by replacing sec²x dx with du and tan¹ᐟ²x with u¹ᐟ².

Step 3: Rewrite the integral in terms of u: ∫ sec²x tan¹ᐟ²x dx becomes ∫ u¹ᐟ² du.

Step 4: Apply the power rule for integration: ∫ uⁿ du = (uⁿ⁺¹)/(n+1), where n ≠ -1. Here, n = 1/2, so the integral becomes (u³ᐟ²)/(3/2) + C.

Step 5: Substitute back u = tan(x) to express the result in terms of x: (tan³ᐟ²x)/(3/2) + C.

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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 secant (sec) and tangent (tan), are fundamental in calculus, representing relationships between angles and sides of triangles. The secant function is the reciprocal of the cosine function, while the tangent function is the ratio of the sine to the cosine. Understanding these functions is crucial for evaluating integrals involving them.

Integration Techniques

Integration techniques, including substitution and integration by parts, are essential for solving complex integrals. In the case of the integral ∫ sec²x tan¹ᐟ²x dx, recognizing the relationship between the functions can simplify the process. Mastery of these techniques allows for the effective evaluation of integrals that may not be straightforward.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, stating that if a function is continuous on an interval, the integral of its derivative over that interval equals the difference in the function's values at the endpoints. This theorem provides a foundation for evaluating definite and indefinite integrals, making it a key concept in understanding the behavior of functions and their integrals.