(2^ln x)/x Antiderivative Example

Khan Academy
22 Oct 200908:40
EducationalLearning
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TLDRThe video script discusses the process of solving the indefinite integral of 2 to the power of the natural log of x, divided by x. The solution involves u-substitution, where u is defined as the natural log of x. By rewriting 2 as e to the natural log of 2 and applying exponent rules, the integral is simplified to a form that allows for easy integration. The final antiderivative is expressed as 1 over the natural log of 2 times 2 to the natural log of x, plus a constant c. The explanation is detailed, providing clarity on each step of the integration process.

Takeaways
  • 📚 The problem discussed is the indefinite integral of 2 to the power of the natural log of x over x, dx.
  • 🤔 The first step is to recognize the integral's structure and consider u-substitution as a solution method.
  • 🔄 U-substitution is used with u defined as the natural log of x, and du/dx is 1/x, leading to du = (1/x) dx.
  • 📈 After substitution, the integral simplifies to the antiderivative of e to the u times du.
  • 🌟 The integral simplifies further by recognizing that 2 can be written as e to the natural log of 2.
  • 🧠 The problem-solving process involves combining exponent rules and logarithmic properties to manage the complex expression.
  • 🔧 The antiderivative is found to be (1 / (natural log of 2)) * e to the (natural log of 2 * u) + c.
  • 🔄 Reverse substitution is applied by replacing u with the natural log of x to get the final antiderivative in terms of x.
  • 🎯 The final answer is (1 / (natural log of 2)) * 2 to the (natural log of x) + c, showcasing the power of substitution and simplification.
  • 👍 The video is a tribute to the problem posted by Bud on the Khan Academy Facebook page and emphasizes the general interest and educational value of such problems.
Q & A

  • What is the integral problem discussed in the transcript?

    -The integral problem discussed is finding the indefinite integral of 2 to the power of the natural log of x, divided by x, dx.

  • What is the first step suggested when encountering this type of integral?

    -The first step is to recognize that the integral can be rewritten as the integral of one over x times 2 to the power of the natural log of x dx.

  • Why is u-substitution used in this problem?

    -U-substitution is used because the derivative of the natural log of x, which is 1/x, allows for a simpler integration process.

  • What is the substitution defined in the script?

    -The substitution defined is u = natural log of x, and du/dx = 1/x or du = 1/x dx.

  • How does the script transform the number 2 in the integral?

    -The number 2 is transformed to e to the power of the natural log of 2, based on the definition of the natural log.

  • What property of exponents is used to simplify the integral further?

    -The property that raising a number to an exponent and then to another exponent is the same as raising the base to the product of those exponents is used.

  • What is the antiderivative of e to the au du?

    -The antiderivative is 1 over a times e to the au, plus a constant c, derived from the fundamental theorem of calculus and the chain rule.

  • How is the antiderivative expressed in terms of the original variable x after substitution?

    -After reverse substitution, the antiderivative is expressed as 1 over the natural log of 2 times 2 to the natural log of x plus a constant c.

  • What simplification is done at the end of the script?

    -The simplification involves using the property that a times the natural log of b is equal to the natural log of b to the power of a, which allows the expression to be rewritten as 2 to the u, and ultimately as 2 to the natural log of x.

  • What is the final answer to the integral problem?

    -The final answer to the integral problem is 1 over the natural log of 2 times 2 to the natural log of x plus a constant c.

  • How does the script demonstrate the use of logarithm and exponent properties in solving integrals?

    -The script demonstrates the use of logarithm and exponent properties by transforming the integral into a more manageable form through substitution and simplification, ultimately leading to the solution.

Outlines
00:00
📚 Introduction to the Integral Problem

The paragraph begins with the speaker discussing a problem posted on the Khan Academy Facebook page by Bud Denny, which involves solving the indefinite integral of 2 to the power of the natural log of x, over x dx. The speaker acknowledges that although Abhi Khanna has provided a correct solution on the discussion board, they wish to create a video to explain the solution in more detail due to its general interest. The speaker introduces the problem and suggests using substitution as the method to tackle the integral, recognizing the natural log of x in the numerator as a clue to start with. The speaker then proceeds to explain the u-substitution technique, setting u as the natural log of x, and du/dx as 1/x, which simplifies the integral to the antiderivative of 2 to the u times du.

05:03
🔍 Solving the Integral Using Exponential Properties

In this paragraph, the speaker continues to work on the integral problem by transforming the expression to make it more manageable. They recognize that the base 2 can be rewritten as e to the natural log of 2, which allows them to apply the concept of exponent rules more effectively. The speaker then simplifies the expression by combining exponents and applying the properties of logarithms to rewrite the antiderivative as a product of constants and exponential functions. The speaker also explains the process of reverse substitution, replacing u with the natural log of x to find the original antiderivative with respect to x. The final result of the integral is presented as 1 over the natural log of 2 times 2 to the natural log of x plus a constant (c). The speaker emphasizes the satisfaction of simplifying a complex-looking antiderivative problem and credits Bud for posting the problem.

Mindmap
Keywords
💡indefinite integral
The indefinite integral refers to the antiderivative, which is a function whose derivative is the given function. In the context of the video, the indefinite integral is the mathematical operation being sought for the function 2 to the natural log of x over x. The process of finding the indefinite integral involves using integration techniques such as substitution and properties of logarithms and exponents.
💡natural log
The natural log, often denoted as ln(x), is the logarithm of a number with base 'e', where 'e' is the mathematical constant approximately equal to 2.71828. It is the inverse operation of exponentiation with base 'e'. In the video, the natural log is used as part of the integrand, which is the function being integrated.
💡u-substitution
U-substitution is a method used in integration to simplify the process by replacing a part of the integrand with a new variable, 'u'. This technique is particularly useful when the derivative of the integrand is easily recognizable or when the integrand can be expressed in terms of its derivative. In the video, u-substitution is used to transform the integral of 2 to the natural log of x over x into a more manageable form by setting u equal to the natural log of x.
💡derivative
A derivative is a concept in calculus that represents the rate of change or slope of a function at a given point. It is used to find the derivative of a function with respect to a variable, which in the context of integration, helps to determine the antiderivative. In the video, the derivative of the natural log of x, which is 1/x, is used to facilitate the u-substitution method for integration.
💡exponent
An exponent is a mathematical notation that indicates the number of times a base is multiplied by itself. In the context of the video, the concept of exponents is crucial in transforming the integral expression and simplifying it through properties of exponents and natural logs. The video discusses raising a number (e) to the power of a variable (u) and how this relates to the function being integrated.
💡chain rule
The chain rule is a fundamental calculus technique used to find the derivative of a composite function. It states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. In the video, the chain rule is alluded to when discussing the properties of derivatives and antiderivatives, particularly when simplifying the expression involving exponents.
💡antiderivative
An antiderivative, also known as an indefinite integral, is a function that represents the area under the curve of the original function from a starting point to 'x'. It is the reverse process of differentiation and is used to find the integral of a given function. In the video, the antiderivative of the function 2 to the natural log of x over x is the main goal of the problem-solving process.
💡substitution
Substitution is a technique used in calculus to simplify complex integrals by replacing a part of the integrand with another variable. This method often makes the integral easier to solve by converting it into a form that can be more readily integrated. In the video, substitution is a key step in solving the integral by setting u as the natural log of x, which simplifies the process of finding the antiderivative.
💡integration
Integration is a fundamental process in calculus that involves finding the area under a curve or the accumulated quantity represented by a function over an interval. It is the inverse operation of differentiation and is used to find the original function given its derivative. In the video, integration is the main focus as the problem involves finding the indefinite integral of a complex function.
💡constant
In mathematics, a constant is a value that does not change. In the context of the video, constants are used in the process of integration and can be factors in the integrand or appear as coefficients when simplifying the antiderivative. The video mentions a constant (the natural log of 2) which is used to express the number 2 as e to the natural log of 2.
💡properties of logarithms
Properties of logarithms are rules that help simplify and manipulate expressions involving logarithms. These properties include product, quotient, power, and change of base rules. In the video, properties of logarithms are used to transform and simplify the expression 2 to the natural log of x, ultimately leading to the solution of the integral.
Highlights

The problem discussed is the indefinite integral of 2 to the power of the natural log of x over x, dx.

The first step in solving the integral is to recognize the natural log of x in the numerator and consider how to proceed.

The integral can be rewritten as the integral of one over x times 2 to the power of the natural log of x dx.

The derivative of the natural log of x with respect to x is equal to 1/x, which is crucial for the substitution method.

The substitution u = natural log of x and du dx = 1/x dx is used to simplify the integral.

After substitution, the integral simplifies to the indefinite integral of e to the u times the natural log of 2 du.

The integral simplifies further by recognizing that 2 is e to the natural log of 2.

The antiderivative of e to the au du is 1 over a e to the au plus a constant c, which is a key step in solving the integral.

The antiderivative is expressed as 1 over the natural log of 2 times e to the u times the natural log of 2, plus a constant c.

The natural log of 2 to the u is the power you raise e to get 2 to the u, which simplifies the expression further.

The final antiderivative is 1 over the natural log of 2 times 2 to the natural log of x plus a constant c, after reverse substitution.

The problem showcases the power of substitution and logarithmic properties in simplifying complex integrals.

The solution involves creative manipulation of exponents and logarithms to arrive at a tractable form.

The problem and its solution are of general interest due to the innovative approach to integral calculus.

The discussion board on the Khan Academy Facebook page is a platform where such mathematical problems are shared and solved.

Abhi Khanna provided a correct solution on the discussion board, which prompted the video explanation.

The video aims to provide a quick explanation of the solution for the benefit of those interested in the problem.

Transcripts

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