Integration Practice II | MIT 18.01SC Single Variable Calculus, Fall 2010

MIT OpenCourseWare
7 Jan 201114:46
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TLDRIn this recitation video, the instructor guides through solving two integration problems: one definite integral of secant cubed u from -Ο€/4 to Ο€/4, and one indefinite integral for the antiderivative of 1/(2 sin u cos u). The instructor explains breaking down the integrals, using integration by parts for the first problem, and utilizing trigonometric identities for the second. Emphasis is placed on the importance of knowing trigonometric identities and their application in simplifying complex integrals. The session concludes with a detailed step-by-step solution to both problems.

Takeaways
  • πŸ“š The video focuses on practicing integration with two given problems: a definite integral and an indefinite integral.
  • πŸ” The first problem involves integrating secant cubed of 'u' from -Ο€/4 to Ο€/4, which requires breaking down the integral and using integration by parts.
  • πŸ“ The secant cubed integral is split into secant 'u' and secant squared 'u', with the latter being transformed into a form involving tangent squared 'u'.
  • πŸ”‘ The identity secant squared 'u' equals 1 plus tangent squared 'u' is used to simplify the integral.
  • 🚫 The video explains that a direct substitution is not possible due to the nature of the derivatives of secant and tangent functions.
  • πŸ”„ Integration by parts is then used to tackle the integral of secant 'u' tangent squared 'u', leading to a recursive form of the integral.
  • πŸ” The solution involves recognizing the recursive nature and solving for the integral by moving terms to opposite sides and dividing by 2.
  • πŸ“ The antiderivative of secant 'u' is given as 1/2 natural log of the absolute value of secant 'u' plus tangent 'u'.
  • πŸ“Œ The bounds are applied at the end to evaluate the definite integral, using the values of secant and tangent at Ο€/4 and -Ο€/4.
  • πŸ“ˆ The second problem is an indefinite integral of 1/2 sine 'u' cosine 'u', which simplifies to the integral of du over sine 2'u' using a trigonometric identity.
  • πŸ“˜ The antiderivative of cosecant 'u' is applied, adjusted for the factor of 2 in the argument, resulting in -1/2 natural log of the absolute value of cosecant 2'u' plus cotangent 2'u'.
  • 🧠 The importance of knowing basic trigonometric identities for simplifying integral problems is emphasized.
Q & A
  • What are the two types of integration problems discussed in the video?

    -The video discusses a definite integral and an indefinite integral problem.

  • What is the definite integral problem in the video?

    -The definite integral problem is to find the value of the integral from -Ο€/4 to Ο€/4 of secant cubed u du.

  • What is the indefinite integral problem in the video?

    -The indefinite integral problem is to find an antiderivative for the function 1/2 sine u times cosine u.

  • Why does the instructor suggest splitting the secant cubed integral into two parts?

    -The instructor suggests splitting the integral into secant u and secant squared u to simplify the problem and make it easier to integrate.

  • What trigonometric identity is used to simplify the indefinite integral problem?

    -The trigonometric identity 2 sine u times cosine u equals sine 2u is used to simplify the indefinite integral problem.

  • Why does the instructor decide not to use substitution for the integral of secant u times tangent squared u?

    -The instructor decides not to use substitution because the derivative of tangent is secant squared and the derivative of secant is secant tangent, which does not match the form needed for a simple substitution.

  • What strategy is used instead of substitution for the integral of secant u times tangent squared u?

    -Integration by parts is used instead of substitution to solve the integral of secant u times tangent squared u.

  • How does the instructor handle the integral of secant cubed u to simplify it?

    -The instructor uses integration by parts and then rearranges the equation to isolate the integral of secant cubed u, which simplifies the problem.

  • What is the antiderivative of secant u?

    -The antiderivative of secant u is 1/2 natural log of the absolute value of secant u plus tangent u.

  • How does the instructor evaluate the definite integral with the bounds -Ο€/4 to Ο€/4?

    -The instructor evaluates the definite integral by plugging in the bounds into the simplified antiderivative expression and calculating the difference.

  • What is the final result of the definite integral from -Ο€/4 to Ο€/4 of secant cubed u du?

    -The final result is √2 + 1/2 natural log of the absolute value of (√2 + 1)/(√2 - 1).

  • What is the antiderivative of 1/2 sine u times cosine u found in the video?

    -The antiderivative is -1/2 natural log of the absolute value of cosecant 2u plus cotangent 2u.

  • Why is the chain rule important when finding the antiderivative of cosecant 2u?

    -The chain rule is important because it accounts for the extra factor of 2 in the derivative of cosecant 2u, which is then canceled out by the 1/2 in the antiderivative.

Outlines
00:00
πŸ“š Introduction to Integration Practice

The video script begins with an introduction to a session on integration practice. The instructor plans to tackle two integration problems: a definite integral involving the function secant cubed and an indefinite integral of a product of sine and cosine functions. The audience is encouraged to pause the video to attempt the problems before the instructor demonstrates the solutions. The focus is on finding the value of the definite integral from -Ο€/4 to Ο€/4 of secant cubed of u, and finding an antiderivative for the product of 1/2, sine of u, and cosine of u.

05:03
πŸ” Detailed Solution for Secant Cubed Integral

The instructor delves into solving the definite integral of secant cubed of u. Initially, the bounds are omitted for simplicity, and the integral is split into secant u and secant squared u. The strategy involves recognizing secant squared u as 1 plus tangent squared u, which allows for partial integration. The integral of secant u is straightforward, but the integral of secant u times tangent squared u requires integration by parts. The instructor chooses secant u tangent u as the function for integration by parts, leading to a recursive integral form that simplifies after some algebraic manipulation. The solution involves recognizing the antiderivative of secant u and applying bounds to find the final value of the integral.

10:06
πŸ“ˆ Evaluating the Definite Integral with Bounds

After establishing the antiderivative involving natural logarithms, the instructor proceeds to evaluate the definite integral by applying the bounds of -Ο€/4 and Ο€/4. The secant and tangent values at these bounds are calculated, considering their even and odd properties, respectively. The final evaluation involves plugging in these trigonometric values into the antiderivative formula and simplifying the expression to arrive at the integral's value.

πŸ“˜ Antiderivative of Sine and Cosine Product Using Trigonometric Identities

The script concludes with finding an antiderivative for the product of 1/2, sine of u, and cosine of u. The instructor emphasizes the utility of knowing basic trigonometric identities, specifically the double angle formula, which simplifies the integral to one over sine of 2u. The antiderivative of cosecant is then applied, with adjustments for the factor of 2 in the argument, resulting in a final expression involving natural logarithms of the absolute values of cosecant and cotangent of 2u, each divided by 2.

Mindmap
Keywords
πŸ’‘Integration
Integration is a fundamental concept in calculus that represents the process of finding a quantity given its rate of change. In the context of the video, integration is used to calculate the area under a curve or to find an antiderivative. The script discusses both definite and indefinite integration problems, emphasizing the process of finding antiderivatives for given functions.
πŸ’‘Definite Integral
A definite integral specifies the exact limits between which the integration is performed. It is used to calculate the area under a curve between two points. In the video, the instructor is focusing on a definite integral from -Ο€/4 to Ο€/4 of secant cubed u, demonstrating the steps to find the exact value of the integral.
πŸ’‘Indefinite Integral
An indefinite integral is an antiderivative that does not have specific limits and is often denoted with an 'S' notation. It represents a family of functions that differ by a constant. The video script includes an indefinite integral of 1/2 sine u cosine u, where the goal is to find the antiderivative without applying specific bounds.
πŸ’‘Secant
Secant is a trigonometric function defined as the reciprocal of the cosine function, sec(x) = 1/cos(x). In the video, the secant function is raised to the power of three, and the integral of secant cubed u is a central focus. The script discusses strategies for integrating this function, including splitting it into simpler components.
πŸ’‘Tangent
Tangent is a trigonometric function that represents the ratio of the opposite side to the adjacent side in a right-angled triangle, tan(x) = sin(x)/cos(x). The video mentions tangent in the context of integrating secant u times tangent squared u, where understanding the derivative of tangent is crucial for the integration process.
πŸ’‘Integration by Parts
Integration by Parts is a technique used in calculus to integrate products of functions. It is analogous to the product rule for differentiation, and the formula is ∫u dv = uv - ∫v du. In the script, the instructor uses integration by parts to tackle the integral of secant u tangent squared u, which is a key step in solving the problem.
πŸ’‘Trigonometric Identities
Trigonometric identities are equations that relate different trigonometric functions to one another. They are essential for simplifying and solving trigonometric integrals. The video script highlights the identity 2 sine u cosine u = sine 2u, which simplifies the integral of 1/2 sine u cosine u into a more manageable form.
πŸ’‘Cosecant
Cosecant is the reciprocal of the sine function, csc(x) = 1/sin(x). It is used in the video to transform the integral of 1/2 sine u cosine u into an integral involving cosecant, which is then integrated using knowledge of the antiderivative of cosecant.
πŸ’‘Chain Rule
The Chain Rule is a fundamental principle in calculus for differentiating composite functions. In the context of integration, it helps to adjust the antiderivative when the function inside the integral is a composite function. The script uses the chain rule to adjust the antiderivative of cosecant when the argument is 2u.
πŸ’‘Natural Logarithm
The natural logarithm is the logarithm to the base e, commonly denoted as ln(x). It appears in the video as part of the antiderivative for the function involving cosecant. The script uses the natural logarithm to express the antiderivative of the transformed integral and to evaluate the definite integral.
Highlights

Introduction to the video with a focus on practicing integration with two problems: a definite integral and an indefinite integral.

The first problem involves finding the definite integral of secant cubed u from -Ο€/4 to Ο€/4.

The second problem is to find an antiderivative for 1/2 sine u times cosine u.

A suggestion to pause the video and attempt the problems before continuing to watch the solution.

Starting the solution by dropping the bounds to simplify the process of finding the antiderivative.

The approach to split the integral of secant cubed u into secant u and secant squared u.

Using the identity secant squared u equals 1 plus tangent squared u to simplify the integral.

Recognizing that a direct substitution is not possible due to the derivatives of secant and tangent.

Deciding to use integration by parts to handle the integral of secant u times tangent squared u.

Choosing secant u tangent u as the function for integration by parts and explaining the selection.

Using integration by parts formula to set up the integral and anticipate the resulting secant cubed u.

Manipulating the integral to isolate secant cubed u and prepare for solving it.

Dividing the integral by 2 to simplify the expression and find the antiderivative.

Identifying the antiderivative of secant u and applying it to the problem.

Evaluating the antiderivative at the given bounds and using the values of secant and tangent at Ο€/4 and -Ο€/4.

Using trigonometric identities to simplify the second integral into an integral over sine 2u.

Applying the double angle formula to transform the integral into an expression involving cosecant.

Finding the antiderivative of cosecant 2u by adjusting for the chain rule and the presence of the factor 2.

Emphasizing the importance of knowing basic trigonometric identities for simplifying integral problems.

Concluding the video by summarizing the method used for both integrals and the importance of integration by parts and trigonometric identities.

Transcripts
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