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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1:14 minutes

Problem 4.R.95

Textbook Question

90–103. Indefinite integrals Determine the following indefinite integrals.

∫(1 + 3 cosΘ) dΘ

Verified step by step guidance

Step 1: Recognize that the integral is in the form ∫(1 + 3 cosΘ) dΘ, which can be split into two separate integrals: ∫1 dΘ + ∫3 cosΘ dΘ.

Step 2: Solve the first integral ∫1 dΘ. The integral of 1 with respect to Θ is simply Θ, because the derivative of Θ is 1.

Step 3: Solve the second integral ∫3 cosΘ dΘ. Use the constant multiple rule, which allows you to factor out the constant 3, resulting in 3∫cosΘ dΘ.

Step 4: Recall the integral of cosΘ with respect to Θ is sinΘ. Therefore, ∫cosΘ dΘ = sinΘ, and 3∫cosΘ dΘ = 3sinΘ.

Step 5: Combine the results from Step 2 and Step 4. The indefinite integral ∫(1 + 3 cosΘ) dΘ is Θ + 3sinΘ + C, where C is the constant of integration.

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Key Concepts

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

Indefinite Integral

An indefinite integral represents a family of functions whose derivative is the integrand. It is expressed without limits of integration and includes a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antidifferentiation.

Basic Integration Rules

Basic integration rules are fundamental formulas used to compute integrals. For example, the integral of a constant 'a' is 'aΘ', and the integral of cos(Θ) is sin(Θ). Understanding these rules is essential for solving integrals efficiently and accurately.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that arise in various mathematical contexts, including calculus. Their properties, such as periodicity and symmetry, play a crucial role in integration, especially when dealing with integrals involving trigonometric expressions.