Trig Integral Practice | MIT 18.01SC Single Variable Calculus, Fall 2010

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7 Jan 201111:22
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TLDRIn this recitation, Joel Lewis revisits integrals involving trigonometric functions, offering three additional examples to supplement Professor Miller's lecture. He emphasizes the importance of trigonometric identities, particularly the angle sum identity for cosine, for solving these integrals. Lewis demonstrates the integration process for expressions like sine cubed x secant squared x, using substitutions and trigonometric identities to simplify the calculations. He also tackles more complex integrals involving multiple trigonometric functions with different arguments, such as sine x cosine of 2x and sine 2x cosine 3x, guiding students through the process of simplifying and integrating these expressions.

Takeaways
  • 📚 The video is a continuation of a lecture series on integrals involving trigonometric functions.
  • 🔍 The focus is on three additional integral examples that require the use of trigonometric identities.
  • 🧠 The angle sum identity for cosine is essential for solving one of the integrals: cos(a + b) = cos a cos b - sin a sin b.
  • 📝 The first integral involves secant squared, which is closely related to the cosine function, and is expressed as 1/cosine.
  • 🤔 The integral of sine cubed x secant squared x is simplified by breaking it down to involve powers of sine and cosine.
  • 📉 The substitution method is used for integration, where u is set to cosine x, simplifying the integral to a form involving u to the power of -2.
  • 🔄 The double angle formula is applied to transform cosine 2x into a form involving only sine and cosine of x.
  • 📈 The integral of sine x times cosine 2x is simplified using the double angle formula and then solved using substitution.
  • 📌 The third integral involves sine 2x and cosine 3x, which are both transformed using double angle and sum formulas to simplify the expression.
  • ✍️ The final answer for the third integral is given as a combination of cosine terms with different powers, resulting in a constant of integration.
  • 📝 The script encourages students to pause the video to work through the integrals and check their answers against the provided solutions.
Q & A

  • What is the main focus of the recitation session led by Joel Lewis?

    -The main focus of the recitation session is to provide additional examples of integrals involving trigonometric functions, building on what was covered in Professor Miller's lecture.

  • Which trigonometric identity is highlighted as important for solving one of the integrals?

    -The angle sum identity for cosine, which states that cos(a + b) = cos(a)cos(b) - sin(a)sin(b), is highlighted as important for solving one of the integrals.

  • How does Joel Lewis suggest handling integrals where sine appears with an odd power?

    -Joel Lewis suggests breaking up the sine term so that it includes one instance of sine and converting the remaining sine squared terms into 1 - cos^2. This simplifies the integral by reducing it to a form involving sine and cosine.

  • What substitution does Joel Lewis use for the first integral involving sine cubed x secant squared x?

    -For the first integral, Joel Lewis sets u equal to cos(x), which simplifies the integral by converting sine terms to du and cosine terms to u.

  • What is the result of the first integral after substitution and simplification?

    -The result of the first integral is sec(x) + cos(x) + C.

  • What double angle formula does Joel Lewis use to simplify the second integral involving sine x cosine 2x?

    -Joel Lewis uses the double angle formula for cosine, cos(2x) = 2cos^2(x) - 1, to simplify the second integral.

  • What is the simplified result for the second integral involving sine x cosine 2x?

    -The simplified result for the second integral is -2/3 cos^3(x) + cos(x) + C.

  • How does Joel Lewis handle the integral involving sine 2x cosine 3x?

    -Joel Lewis handles the integral involving sine 2x cosine 3x by expressing both functions in terms of sine and cosine of x using angle sum and double angle formulas, then simplifying.

  • What is the result of the third integral after simplification?

    -The result of the third integral is -8/5 cos^5(x) + 2 cos^3(x) + C.

  • Why is it important to write everything in terms of sine x and cosine x for the integrals discussed?

    -It is important to write everything in terms of sine x and cosine x to make the integrals easier to solve using substitutions and familiar trigonometric identities.

Outlines
00:00
📚 Introduction to Trigonometric Integrals

Joel Lewis begins the recitation by welcoming students back and referencing the lecture on integrals involving trigonometric functions by Professor Miller. He introduces three additional integral examples that students should attempt, emphasizing the importance of recalling trigonometric identities, specifically the angle sum identity for cosine. Lewis encourages students to pause the video to work through the problems and then return to compare their answers. The first integral discussed is sine cubed x secant squared x, and Lewis explains the relationship between secant and cosine, suggesting a substitution method to simplify the integral.

05:03
🔍 Simplifying Trigonometric Integrals with Substitutions

The second paragraph delves into the process of simplifying the integral of sine x times cosine of 2x. Lewis explains that the integral involves a trigonometric function of 2x, necessitating the use of double angle formulas to express the integral in terms of sine and cosine of x. He outlines the steps for substitution, using u as a variable to represent cosine x, and simplifies the integral to a form that can be easily integrated. The summary includes the application of double angle formulas and the final result of the integral.

10:04
📘 Advanced Trigonometric Integral Techniques

In the third paragraph, Lewis tackles the more complex integral involving sine 2x and cosine 3x. He discusses the importance of having matching arguments in trigonometric integrals and uses the angle sum identity to express cosine 3x in terms of sine and cosine of x. Lewis then applies the double angle formula to sine 2x and simplifies the integral to a form involving powers of cosine and a single power of sine. He suggests a substitution method for simplification but does not complete the process, instead providing the final answer for students to verify their work.

Mindmap
Keywords
💡Trigonometric functions
Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. In the context of the video, these functions are integral to solving calculus problems involving integrals, such as sine and cosine, which are the most commonly used trigonometric functions. The script discusses integrals involving trigonometric functions, emphasizing their importance in solving the given examples.
💡Integrals
Integrals in calculus represent the process of finding the area under a curve defined by a function between two points. In the video, the main theme revolves around computing integrals of trigonometric functions, which are a type of mathematical problem that requires knowledge of trigonometric identities and calculus techniques.
💡Trig identities
Trigonometric identities are equations that hold true for all values of the variables involved. They are essential tools in solving integrals involving trigonometric functions. The script mentions the need to remember trig identities, particularly the angle sum identity for cosine, to compute the integrals presented.
💡Angle sum identity
The angle sum identity for cosine is a specific trigonometric identity that states that the cosine of the sum of two angles is equal to the product of the cosines of the individual angles minus the product of their sines. In the script, this identity is highlighted as necessary for solving one of the integrals, showing its relevance in advanced calculus problems.
💡Secant function
The secant function is the reciprocal of the cosine function, often denoted as sec(x) = 1/cos(x). It is closely related to the cosine function and appears in the first integral example in the script, where the讲师relates it to the integral of powers of sine and cosine, illustrating its role in solving integrals involving trigonometric functions.
💡Odd power
In the context of the script, an odd power refers to an exponent that is an odd integer. When a trigonometric function, like sine, is raised to an odd power, it simplifies the process of integration, as shown in the first integral example where sine cubed is broken down using trigonometric identities.
💡Substitution
Substitution is a common technique used in calculus, particularly when integrating, to simplify the expression by replacing a part of the integral with a new variable. In the script, the讲师uses substitution by setting u equal to cosine x to simplify the integral and make it easier to solve.
💡Double angle formula
The double angle formula is a trigonometric identity that expresses the sine or cosine of twice an angle in terms of the sine and cosine of the original angle. In the script, the讲师uses the double angle formula to expand cosine 2x and simplify the integral of sine x times cosine of 2x.
💡Cosine sum formula
The cosine sum formula is another trigonometric identity used to express the cosine of the sum of two angles. The script mentions using the cosine sum formula to express cosine 3x in terms of cosine x and sine x, which is crucial for simplifying the integral of sine 2x times cosine 3x.
💡Integration by parts
Although not explicitly mentioned in the script, the process of solving the integrals described involves a form of integration by parts, especially when dealing with products of trigonometric functions. The讲师demonstrates this concept by breaking down the integrals into simpler forms that can be integrated using known techniques.
💡Constant of integration
The constant of integration is added when finding the indefinite integral of a function to account for the infinite number of possible original functions that could have produced the given antiderivative. In the script, after solving the integrals, the讲师includes a constant of integration to indicate the general solution to the integral problems.
Highlights

Introduction to integrating trigonometric functions, focusing on examples different from those in the lecture.

Reminder of the cosine angle sum identity: cos(a + b) = cos(a)cos(b) - sin(a)sin(b).

First integral example: Integrating sine cubed x secant squared x.

Explanation of secant as 1 over cosine and transforming the integral into a power of sine and cosine.

Strategy for integrals with odd powers of sine: Break down sine cubed x into sine x times sine squared x.

Substitute sine squared x with 1 - cosine squared x to simplify the integral.

Introduction of u-substitution: Let u = cos(x) and du = -sin(x)dx.

First integral solution: Resulting in sec(x) + cos(x) + C.

Second integral example: Integrating sine x cosine 2x.

Use of double angle formula for cosine: cos(2x) = 2cos^2(x) - 1.

Simplification using substitution u = cos(x), leading to the integral of 2cos^2(x)sin(x).

Second integral solution: Resulting in -2/3 cos^3(x) + cos(x) + C.

Third integral example: Integrating sine 2x cosine 3x.

Application of angle sum formula and double angle formula to express cos(3x) and sin(2x) in terms of sine and cosine.

Third integral solution: Resulting in -8/5 cos^5(x) + 2 cos^3(x) + C.

Transcripts

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