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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1:28 minutes

Problem 4.R.97

Textbook Question

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

∫ dx / (1 - sin² x)

Verified step by step guidance

Recognize that the denominator, \(1 - \sin^2 x\), can be simplified using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). This simplifies \(1 - \sin^2 x\) to \(\cos^2 x\).

Rewrite the integral as \(\int \frac{dx}{\cos^2 x}\).

Recall that \(\frac{1}{\cos^2 x}\) is equivalent to \(\sec^2 x\). Substitute this into the integral to get \(\int \sec^2 x \, dx\).

Use the standard integral formula \(\int \sec^2 x \, dx = \tan x + C\), where \(C\) is the constant of integration.

Conclude that the solution to the integral is \(\tan x + C\).

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Key Concepts

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

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antidifferentiation, and it is fundamental in calculus for solving problems related to area under curves and accumulation functions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. A key identity relevant to the given integral is the Pythagorean identity, which states that sin²(x) + cos²(x) = 1. This identity can be used to simplify expressions involving sine and cosine, making it easier to evaluate integrals that include these functions.

Substitution Method

The substitution method is a technique used in integration to simplify the integrand by changing variables. This method involves substituting a part of the integrand with a new variable, which can make the integral easier to solve. In the context of the given integral, recognizing that 1 - sin²(x) can be rewritten as cos²(x) allows for a straightforward integration process.