Integration by parts: º___cos(x)dx | AP Calculus BC | Khan Academy
TLDRThe video script discusses the process of finding the antiderivative of e^x*cos(x) using integration by parts. It explains the selection of functions for u and dv, and the subsequent steps to apply the integration by parts formula. The script also highlights the challenge of dealing with the sine function that arises and how to address it by reassigning functions and applying integration by parts again. Ultimately, it leads to the solution, which includes an antiderivative with a constant term, emphasizing the importance of including the constant when solving integrals.
Takeaways
- 📚 The problem discussed is finding the antiderivative of e^(x)cos(x)dx using integration by parts.
- 🤔 Neither e^(x) nor cos(x) becomes significantly simpler or more complicated when taking derivatives or antiderivatives, making the choice of f(x) and g'(x) somewhat arbitrary.
- 🌟 The initial choice assigns f(x) = e^(x) and g'(x) = cos(x), with the antiderivative of g(x) being sin(x).
- 🔄 Integration by parts is applied, resulting in an expression involving e^(x)sin(x) and the antiderivative of e^(x)sin(x).
- 🧐 The antiderivative of e^(x)sin(x) is found by reapplying integration by parts, leading to an expression involving e^(x)cos(x) and an antiderivative of negative cos(x) which is sin(x).
- 🔄 A substitution step is performed, replacing the derived expression from the second integration by parts into the original equation.
- 📈 By adding the integral of e^(x)cos(x) to both sides of the equation, the original expression is isolated and can be solved for.
- 🎯 The antiderivative of e^(x)cos(x)dx is found to be e^(x)sin(x) + e^(x)cos(x)/2 + C, where C is the constant of integration.
- 📊 The process involves understanding the concept of integration by parts and the importance of including the constant of integration in the final antiderivative.
- 🌐 The solution process highlights the utility of substitution and back-solving in tackling complex integrals.
Q & A
What integration technique is discussed in the script?
-The script discusses the use of integration by parts to find the antiderivative of a given function.
Which two functions are considered for integration by parts in the script?
-The two functions considered are e to the x (exponential function) and cosine of x.
Why is it a toss-up which function to assign to f(x) and which to g'(x) in this case?
-It's a toss-up because neither of the functions, e to the x nor cosine of x, becomes significantly simpler or more complicated when taking their derivatives or antiderivatives.
How are f(x) and g'(x) assigned in the initial integration by parts?
-f(x) is assigned as e to the x, and g'(x) is assigned as cosine of x.
What is the antiderivative of cosine of x?
-The antiderivative of cosine of x is sine of x.
How does the script suggest solving the indefinite integral involving sine of x?
-The script suggests solving it by reassigning f(x) and g(x) and applying integration by parts again.
What is the result of the second application of integration by parts?
-The result is an expression involving negative e to the x times cosine of x, minus the antiderivative of e to the x times sine of x.
How does the script handle the integral that seems to have not made progress?
-The script handles it by substituting the result from the second integration by parts back into the original equation and solving for the original expression.
What is the final expression found for the antiderivative of e to the x cosine of x dx?
-The final expression is e to the x sine of x plus e to the x cosine of x divided by 2, plus a constant.
Why is a constant included in the final antiderivative expression?
-A constant is included because the antiderivative can have an arbitrary constant term, which is a fundamental property of indefinite integrals.
What is the significance of the final result in terms of the original integral?
-The final result is significant because it provides a neat and simplified expression for the antiderivative of the original function, e to the x cosine of x dx.
Outlines
📚 Integration by Parts for e^x Cos(x) Antiderivative
This paragraph discusses the process of finding the antiderivative of the product of two functions, specifically e^x and cos(x), using integration by parts. It begins by acknowledging the complexity of choosing which function to assign as f(x) and which as g'(x), ultimately deciding on f(x) = e^x and g'(x) = cos(x). The explanation includes the steps of applying integration by parts, substituting the derivatives and antiderivatives of the chosen functions, and simplifying the resulting expression. The paragraph emphasizes the challenge of dealing with the sine function in the indefinite integral and attempts to solve it by reassigning f(x) and g'(x) and applying integration by parts again. The process leads to a seemingly circular result, but the paragraph suggests a creative approach by substituting back the derived expression into the original equation, leading to a cleaner expression involving e^x, sin(x), and cos(x).
🔄 Solving the Integral by Substitution and Isolating the Variable
This paragraph continues the exploration of the antiderivative of the product of e^x and cos(x) by focusing on the creative manipulation of the derived expressions. It highlights the interesting outcome of having the original expression twice in the equation after substitution, suggesting the assignment of this derived expression to a variable. The paragraph describes the process of adding the integral of e^x cos(x) to both sides of the equation, which simplifies the expression and allows for solving the original antiderivative. The summary emphasizes the importance of remembering to include a constant in the final antiderivative, as it is a fundamental part of indefinite integration. The final expression for the antiderivative is presented as e^x sin(x) plus e^x cos(x) divided by 2, with a reminder that the derivative of this expression will yield the original function, e^x cos(x), confirming the correctness of the solution.
Mindmap
Keywords
💡Integration by Parts
💡Antiderivative
💡e^x
💡Cosine of x (cos(x))
💡Sine of x (sin(x))
💡Function Assignment
💡Derivative
💡Negative Cosine of x (-cos(x))
💡Substitution
💡Constant of Integration
Highlights
The discussion begins with the exploration of using integration by parts to find the antiderivative of e^x*cos(x)dx.
The choice of functions for integration by parts is highlighted, where neither function becomes significantly simpler upon differentiation or integration.
The assignment of f(x) as e^x and g'(x) as cos(x) is made for the integration by parts process.
The derivative of e^x is e^x, and the antiderivative of cos(x) is sin(x), which are the basic building blocks for the integration process.
The application of integration by parts results in an expression involving the antiderivative of e^x*sin(x).
The process of reassigning f(x) and g(x) to simplify the antiderivative of e^x*sin(x) is described.
The derivative of sine is cosine, and the antiderivative of cosine is sine, which is used to further simplify the expression.
Integration by parts is applied again, leading to an expression involving e^x*cos(x) and e^x*sin(x).
A creative approach of substituting the derived expression back into the original equation is introduced.
The process of simplifying the equation by canceling out parts of the expression is detailed.
The equation is manipulated to express the original integral in terms of a new variable, showcasing a clever algebraic technique.
The importance of remembering to include a constant in the antiderivative is emphasized.
The final antiderivative expression is derived as e^x*sin(x) + e^x*cos(x)/2, with the reminder of the constant of integration.
The method of integration by parts is demonstrated to be a powerful tool for solving complex integrals.
The transcript showcases the step-by-step process of tackling a challenging mathematical problem, providing a clear and methodical approach.
The solution process highlights the iterative nature of problem-solving in mathematics, where one may need to revisit and refine their approach.
The transcript serves as an educational resource for those learning about integration techniques and their applications.
The discussion emphasizes the importance of understanding the properties of basic trigonometric functions in the context of integration.
Transcripts
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