Integration using completing the square and the derivative of arctan(x) | Khan Academy

Khan Academy
8 May 201805:27
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TLDRThe video script presents a step-by-step guide to finding the indefinite integral of a complex rational function. The process involves simplifying the denominator, completing the square to resemble the derivative of arctan, and employing u-substitution to transform the integral into a form involving arctan. The final result is an expression involving the arctangent function, showcasing the power of integration techniques in solving mathematical problems.

Takeaways
  • 📚 The problem involves finding the indefinite integral of a complex rational function.
  • 🤔 The initial approach may involve trying multiple integration techniques but facing difficulties.
  • 👉 The strategy is to complete the square in the denominator to transform the expression.
  • 📈 By completing the square, the expression is reformulated to resemble the derivative of arctan.
  • 🔢 The first step is to simplify the denominator by factoring out a common coefficient.
  • 🧩 Adding and subtracting the same value (in this case, 9) helps create a perfect square trinomial.
  • 📊 The integral can be rewritten with the perfect square and an additional term, resembling a standard form.
  • 🔄 A u-substitution is introduced to simplify the expression further, making the arctan connection more evident.
  • 📱 The substitution involves setting u as x - 3/2 and du as 1/2 dx, which simplifies the integral.
  • 🤏 After the u-substitution, the integral simplifies to 1/10 times the arctan of u.
  • 🎓 The final result is the indefinite integral expressed as 1/10 times the arctan of (x - 3/2) plus a constant C.
Q & A

  • What is the given integral that the instructor is attempting to solve?

    -The given integral is one over five x squared minus 30x plus 65 dx.

  • What is the first step the instructor suggests in solving this integral?

    -The first step is to simplify the denominator by factoring out a 5 to make the coefficient of the x squared term equal to 1.

  • How does the instructor plan to transform the given integral to make it resemble the derivative of arctan?

    -The instructor plans to complete the square in the denominator to get it in a form that resembles the derivative of arctan.

  • What is the significance of completing the square in this context?

    -Completing the square helps to transform the integral into a form that looks like the derivative of the arctan function, which simplifies the process of integration.

  • What is the role of the coefficient of x in the process of completing the square?

    -The coefficient of x, which is -6 in this case, is used to determine the value to be added and subtracted (half of the coefficient squared) to complete the square.

  • How does the instructor perform u-substitution in this integral?

    -The instructor sets u equal to x minus three over two, with du equal to one half dx, and then substitutes these into the integral to simplify it further.

  • What does the derivative of arctan of u look like?

    -The derivative of arctan of u is one over u squared plus one.

  • What is the final result of the integral after performing the u-substitution and simplification?

    -The final result is 1/10 times the arctan of (x minus three over two) plus a constant C, representing the indefinite integral.

  • Why is it important to include the constant C in the final result?

    -The constant C is included because we are solving an indefinite integral, which represents a family of functions that differ by a constant.

  • How does the instructor ensure that the value of the denominator does not change when completing the square?

    -The instructor adds and subtracts the same value (in this case, 9) to complete the square without changing the value of the denominator.

  • What is the final expression of the integral after the instructor completes the square and performs the u-substitution?

    -The final expression is 1/10 times the arc tangent of (x minus three over two) plus a constant C.

Outlines
00:00
📚 Solving Indefinite Integral with Complete Square Method

This paragraph delves into the process of finding the indefinite integral of a given function. The instructor introduces a complex integral expression and encourages the audience to attempt solving it before revealing the method. The approach taken involves simplifying the denominator by factoring out a number to make the coefficient of the x squared term equal to one. The instructor then employs the technique of completing the square to transform the expression into a form resembling the derivative of the arctangent function. The explanation proceeds with a step-by-step breakdown of the process, including the addition and subtraction of a value to create a perfect square and the subsequent rewriting of the integral to highlight its similarity to the arctangent function. The paragraph concludes with the introduction of a substitution method to further simplify the integral and eventually leads to the solution involving the arctangent function, emphasizing the importance of recognizing the derivative of arctangent in the process.

05:00
🎓 Finalizing the Indefinite Integral with Reverse Substitution

In this paragraph, the solution to the indefinite integral is completed through the use of reverse substitution. The instructor starts by reiterating the integral expression derived from the previous paragraph and proceeds to replace the variable 'u' with the expression 'x - 3/2', which was determined during the substitution process. The final step involves simplifying the integral to its final form, which is the arctangent of '(x - 3)/2' multiplied by 1/10, plus a constant 'C'. This conclusion marks the end of the process, providing the audience with a clear and comprehensive solution to the original integral problem.

Mindmap
Keywords
💡indefinite integral
The indefinite integral is a fundamental concept in calculus that represents the reverse process of differentiation. It is used to find the original function from which a given derivative can be obtained. In the context of the video, the instructor is attempting to find the indefinite integral of a given function, which is the main objective and theme of the lesson.
💡multiple integration techniques
Multiple integration techniques refer to various methods used in calculus to evaluate integrals. These can include substitution, integration by parts, and others. In the video, the instructor suggests that one might initially try multiple techniques but then focuses on completing the square as the method of choice for this particular integral.
💡completing the square
Completing the square is a mathematical technique used to transform a quadratic expression into a perfect square trinomial. This is particularly useful when integrating functions that involve quadratic terms. In the video, the instructor uses this method to simplify the denominator of the integral, making it resemble the derivative of the arctangent function.
💡arctangent
The arctangent, or inverse tangent function, is a trigonometric function that takes a ratio of two numbers and returns the angle whose tangent is that ratio. It is often used in integral calculus to simplify certain types of integrals, as it resembles the derivative of the function being integrated. In the video, the instructor notes that the completed square form of the integral resembles the derivative of arctangent, hinting at the use of u-substitution to solve the integral.
💡u-substitution
U-substitution is a method used in calculus to evaluate integrals by replacing a part of the integrand with a new variable, u. This simplifies the integral and often makes it easier to solve. In the video, the instructor uses u-substitution to transform the integral into a form that is recognizable as the derivative of arctangent, allowing for a straightforward solution.
💡derivative
A derivative in calculus represents the rate of change of a function with respect to its variable. It is a fundamental concept used to analyze the behavior of functions and is closely related to the process of integration. In the context of the video, the instructor is working with the integral of a function and notes that the form of the integral resembles the derivative of the arctangent function.
💡constant of integration
The constant of integration, often denoted as C, is a term that appears in indefinite integrals to account for the fact that there are infinitely many antiderivatives that differ by a constant. It acknowledges that the process of integration is not unique, and different antiderivatives can differ by a constant value. In the video, the instructor reminds viewers to include the constant of integration when stating the final answer.
💡reverse substitution
Reverse substitution is the process of replacing the substituted variable, u, back with the original variable, x, to obtain the final form of the integral in terms of the original variable. It is used after performing a u-substitution to solve the integral. In the video, the instructor uses reverse substitution to express the solution in terms of x after solving the integral in terms of u.
💡quadratic expression
A quadratic expression is a mathematical expression that involves a variable raised to the second power. It is typically in the form of ax squared plus bx plus c, where a, b, and c are constants. In the video, the quadratic expression x squared minus 30x plus 65 is the starting point for the integration process.
💡coefficient
A coefficient is a numerical factor that multiplies a variable or term in an algebraic expression. In the context of the video, the instructor refers to the coefficient of the x squared term, which is initially one in the simplified expression.
💡perfect square
A perfect square is a number or an algebraic expression that is the square of an integer or another algebraic expression. In mathematics, transforming an expression into a perfect square can simplify calculations, especially during the process of integration. In the video, the instructor aims to create a perfect square in the denominator to facilitate the integration process.
Highlights

The problem presented is finding the indefinite integral of a complex rational function.

The approach involves multiple integration techniques and completing the square in the denominator.

The goal is to transform the given expression into a form that resembles the derivative of arctangent.

The first step is to simplify the denominator by factoring out a coefficient.

Completing the square is necessary to transform the expression into a perfect square trinomial.

The process includes adding and subtracting the same value to maintain the integrity of the expression.

The integral is rewritten with a perfect square and an additional term, resembling a form related to arc tangent.

An u-substitution is introduced to simplify the integral and reveal its relation to the derivative of arc tangent.

The substitution involves setting u equal to a specific expression involving x and simplifying the integral accordingly.

The derivative of arc tangent is recognized as a key component in the integral.

The integral is expressed in terms of the arc tangent function with a coefficient and constant.

The reverse substitution is performed to express the final result in terms of the original variable x.

The final answer is the integral as a function of arc tangent with the original variable x.

The process demonstrates a methodical approach to solving complex integrals through algebraic manipulation and substitution.

The video provides a detailed walkthrough of the steps, making it an educational resource for similar problems.

The solution showcases the application of advanced calculus techniques in integral calculus.

Transcripts

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MathematicsIntegrationComplete the SquareU-SubstitutionArctangentEducationalMath TutorialAdvanced MathOnline LearningMathematics Education