Zain does Partial Fractions

Chad Gilliland
30 Jan 201404:32
EducationalLearning
32 Likes 10 Comments

TLDRIn this educational video, Zayn Walker demonstrates the process of integration by partial fractions. Starting with the equation \( \frac{5 - x}{(2x - 1)(x + 1)} \), he factors the denominator and sets up the partial fraction decomposition. By plugging in specific values for x to solve for constants a and b, he finds \( a = 3 \) and \( b = -2 \). Walker then integrates the simplified expression, resulting in \( \frac{3}{2} \ln|2x - 1| - 2\ln|x + 1| \), offering a concise explanation suitable for students preparing for exams.

Takeaways
  • 📝 The video is about teaching the method of integration by partial fractions.
  • 🔍 The first step is to factor the denominator of the given integral equation.
  • 📐 The given integral is of the form (5 - x) / ((2x - 1)(x + 1)) which is factored into (5 - x) / [(x - 1/2)(2x + 1)].
  • 📈 The next step involves setting up the equation for partial fractions: A / (2x - 1) + B / (x + 1).
  • 🔄 Multiply the entire equation by the denominator to simplify and find the values of A and B.
  • 📌 To find B, substitute x = -1 into the simplified equation, resulting in B = -3.
  • 📌 To find A, substitute x = 1/2 into the simplified equation, resulting in A = 3.
  • 📚 After finding A and B, rewrite the integral with the identified constants.
  • 📝 The integral is then rewritten as 3 / (2x - 1) - 2 / (x + 1).
  • 🧮 The final step is to integrate each term separately, resulting in 3/2 * ln|2x - 1| - 2 * ln|x + 1|.
  • 📑 The final answer is left as an indefinite integral, without applying limits for a definite integral.
Q & A

  • What is the first step in solving an integral by partial fractions according to the video?

    -The first step is to factor the denominator of the equation.

  • How is the denominator factored in the example provided?

    -The denominator is factored into (2x - 1)(x + 1).

  • What form does the equation take after factoring the denominator?

    -The equation becomes 5 - x over (2x - 1)(x + 1) equals A over (2x - 1) plus B over (x + 1).

  • What is the next step after rewriting the equation with partial fractions?

    -The next step is to multiply both sides of the equation by the common denominator, (2x - 1)(x + 1).

  • How do you determine the values of A and B?

    -You determine the values of A and B by plugging in values for x that simplify the equations. For example, setting x = -1 to solve for B and x = 1/2 to solve for A.

  • What is the value of B when x = -1?

    -When x = -1, B is determined to be -2.

  • What is the value of A when x = 1/2?

    -When x = 1/2, A is determined to be 3.

  • How is the original integral expressed with the determined values of A and B?

    -The original integral is expressed as the integral of (3/(2x - 1)) - (2/(x + 1)) dx.

  • What is the result of integrating the terms separately?

    -The result of integrating the terms separately is 3/2 ln|2x - 1| - 2 ln|x + 1|.

  • What should be done if the integral is definite?

    -If the integral is definite, you should plug in the upper and lower bounds for x to find the actual value.

Outlines
00:00
📚 Introduction to Partial Fractions Integration

Zayn Walker begins a tutorial on partial fractions integration with a friendly greeting to the audience. He introduces the concept by explaining the importance of factoring the denominator of the given integral equation, which in this case is '5 - x' over '2x - 1' and 'x + 1'. He then demonstrates the setup for the partial fraction decomposition by multiplying the equation by the factored denominator to simplify the process of finding the constants 'a' and 'b'.

Mindmap
Keywords
💡Integration
Integration is a fundamental concept in calculus, referring to the process of finding a function given its derivative. In the video, integration is the main theme, as the instructor is teaching how to integrate a complex rational function using a specific method called partial fractions.
💡Partial Fractions
Partial fractions is a technique used in calculus to decompose a complex rational function into simpler fractions, which are easier to integrate. The video demonstrates this method by breaking down the given integral into simpler components that can be more easily solved.
💡Factor
To factor in mathematics means to express a polynomial as the product of its factors. In the context of the video, the denominator of the integral is factored to prepare it for the partial fraction decomposition process.
💡Equation
An equation in mathematics is a statement that asserts the equality of two expressions. In the script, the equation represents the integral that needs to be solved, and the instructor manipulates this equation to apply the method of partial fractions.
💡Integral
An integral is a mathematical concept that represents the area under a curve, and it is the reverse process of differentiation. The video is focused on solving an integral using the method of partial fractions, as indicated by the script.
💡Multiply
In the context of the video, multiplying is used to eliminate the denominator in the equation, which is a step in the process of setting up the partial fraction decomposition.
💡Cancel Out
To cancel out in mathematics means to eliminate terms by performing operations that result in zero. In the script, terms are canceled out to simplify the equation and make it easier to solve for the coefficients in the partial fractions.
💡Coefficients
Coefficients are numerical factors in a term of an algebraic expression. In the video, the instructor is solving for the coefficients 'a' and 'b' in the partial fraction decomposition of the integral.
💡Plug In
To plug in values means to substitute specific values for a variable in an equation to find an unknown. In the script, the instructor plugs in values for 'x' to solve for the coefficients 'a' and 'b' in the partial fractions.
💡Natural Logarithm
The natural logarithm, often denoted as ln(x), is the logarithm to the base 'e', where 'e' is an irrational constant approximately equal to 2.71828. In the video, after finding the partial fractions, the integral is simplified to include natural logarithms, which are then integrated.
💡Absolute Value
Absolute value of a number is its distance from zero on the number line, regardless of direction, denoted as |x|. In the script, the absolute value is used when integrating the partial fraction to ensure the correct sign is maintained.
Highlights

Introduction to the process of integration by partial fractions.

Writing down the equation and factoring the denominator.

Factored form of the denominator: (5 - x)(2x - 1)(x + 1).

Multiplying the equation by the denominator to simplify.

Setting up the equation for partial fractions: A/(2x - 1) + B/(x + 1).

Multiplying through by the denominator to eliminate fractions.

Rewriting the equation as 5 - x = A(x + 1) + B(2x - 1).

Choosing values for x to solve for A and B.

Solving for B by setting x = -1.

Finding B = -2 from the equation 6 = B(-3).

Solving for A by setting x = 1/2.

Calculating A from the equation 9/2 = A(3/2).

Determining A = 3 from the simplified equation.

Substituting A and B back into the original integral.

Setting up the integral of 3/(2x - 1) - 2/(x + 1).

Integrating the partial fractions to get the final answer.

Final answer: 3/2 ln|2x - 1| - 2 ln|x + 1|.

Explanation that the process is complete and the integral is integratable.

Note on handling definite integrals by plugging in values for x.

Conclusion and encouragement for the test.

Transcripts

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Partial FractionsIntegrationMath TutorialEducational ContentAlgebraic TechniquesZayn WalkerIntegral CalculusMath EducationFraction DecompositionStudy Aid