Eric's Calculus Lecture: Evaluate the Indefinite Integral ∫e^(3x+1)dx

Eric Gorenstein

11 Apr 202003:11

Summary

TLDRThis educational video script demonstrates the process of evaluating the indefinite integral of e to the power of 3x plus 1. The presenter uses a simple substitution method, setting u as 3x plus 1, and then applying the antiderivative rule for exponential functions. The integral simplifies to one-third times e to the power of u, and after substituting back, the final answer is one-third e to the power of 3x plus 1 plus C. The script also emphasizes the importance of the constant of integration and suggests verifying the result by differentiating it back to the original function.

Takeaways

📚 The integral to be evaluated is of the form \(\int e^{3x+1} dx\).
🔍 A simple integration rule is used: \(\int e^u du = e^u + C\), where \(C\) is the constant of integration.
🎯 The substitution method is applied with \(u = 3x + 1\), which simplifies the integral.
🔄 The differential \(dx\) is related to \(du\) by \(du = 3dx\), leading to a simplification of the integral.
📈 The integral is transformed to \(\int e^u \cdot \frac{1}{3} du\) after substitution and adjusting for the differential.
🧮 The integral becomes \(\frac{1}{3}e^u + C\) after applying the integration rule.
🔙 The substitution is reversed to express the result in terms of the original variable \(x\), yielding \(\frac{1}{3}e^{3x+1} + C\).
🔍 The correctness of the integral solution can be verified by differentiation.
💡 The derivative of the solution should match the original function \(e^{3x+1}\), confirming the accuracy of the integration process.

Q & A

What is the integral being evaluated in the transcript?
-The integral being evaluated is the indefinite integral of e to the power of (3x + 1).
What integration rule is mentioned in the transcript?
-The integration rule mentioned is the antiderivative of e to the power of U, which is e to the power of U plus a constant of integration, C.
What substitution is used in the solution process?
-The substitution used is U = 3x + 1, which implies that dU = 3dx.
Why is the factor of 3 important in the substitution process?
-The factor of 3 is important because it accounts for the derivative of U with respect to x, ensuring the correct substitution in the integral.
How is the constant of integration represented in the solution?
-The constant of integration is represented as 'C' at the end of the integral solution.
What does the transcript suggest to do after finding the integral?
-The transcript suggests taking the derivative of the solution to ensure it matches the original function.
What is the final expression for the integral after substitution and simplification?
-The final expression for the integral is one-third times e to the power of (3x + 1) plus C.
Why is it necessary to multiply and divide by 1/3 before making the substitution?
-Multiplying and dividing by 1/3 before substitution is necessary to balance the equation and correctly apply the substitution method.
What is the purpose of the phrase 'don't forget the constant' in the transcript?
-The phrase 'don't forget the constant' serves as a reminder to include the constant of integration, C, in the final answer after finding the antiderivative.
How does the transcript ensure the correctness of the integral solution?
-The transcript ensures the correctness of the integral solution by suggesting to differentiate the solution and check if it returns to the original function.
What is the significance of the phrase 'you can always check this' in the transcript?
-The phrase 'you can always check this' implies that there is a verification step involved in the process, which is to differentiate the found integral to confirm its accuracy.

Outlines

00:00

📚 Calculating the Indefinite Integral of e^(3x+1)

The paragraph explains the process of calculating the indefinite integral of e^(3x+1). The speaker begins by stating that they will use a straightforward integration rule, which is the antiderivative of e^U du, equal to e^U + C. They introduce a substitution where U = 3x + 1, and thus dU = 3dx. To adjust for the dx in the integral, they multiply by 3 and divide by 3, resulting in the integral of e^(3x+1) * 3dx. The substitution leads to 1/3 times the integral of e^u du, which simplifies to 1/3 * e^u + C. The final answer is expressed back in terms of x, yielding 1/3 * e^(3x+1) + C. The speaker emphasizes checking the solution by differentiating it to ensure it matches the original function.

Mindmap

Keywords

💡indefinite integral

The indefinite integral, also known as the antiderivative, is a fundamental concept in calculus that represents the family of functions whose derivative is the given function. In the video, the process of finding the indefinite integral of a function involves reversing the action of differentiation. The script specifically mentions evaluating the integral of 'e to the 3x plus 1,' which is a part of the broader theme of integrating exponential functions.

💡antiderivative

An antiderivative is a function whose derivative is equal to a given function. The term is used in the context of the video to describe the process of finding a function that, when differentiated, yields the original function. The video script illustrates this by showing the steps to find the antiderivative of 'e to the 3x plus 1' using a substitution method.

💡integration rule

An integration rule is a mathematical formula used to find the integral of a function. The video script refers to a specific rule for integrating exponential functions, which states that the integral of 'e to the U' with respect to 'U' equals 'e to the U' plus a constant of integration. This rule is central to the video's demonstration of how to integrate the given function.

💡constant of integration

The constant of integration, often denoted as 'C', is added to the result of an indefinite integral to account for the fact that when integrating, we are finding a family of functions that all have the same derivative. In the video, the script reminds viewers not to forget to include this constant, emphasizing its importance in the process of integration.

💡u-substitution

U-substitution is a technique used in calculus to transform an integral into a simpler one by substituting a part of the integral with a new variable, 'u'. The video script describes using u-substitution by letting 'U' be '3x plus 1', which simplifies the integral and allows for the application of the integration rule for exponential functions.

💡Du

In the context of u-substitution, 'Du' represents the differential of the new variable 'u'. The script mentions 'Du' as being equal to '3 times dx', which is derived from the substitution 'U = 3x + 1'. This step is crucial for transforming the original integral into a form that can be easily integrated.

💡exponential function

An exponential function is a mathematical function of the form 'e to the power of x', where 'e' is the base of the natural logarithm. The video's main theme revolves around integrating such functions, specifically 'e to the 3x plus 1'. The script demonstrates how to handle the integration of an exponential function by applying a specific rule and u-substitution.

💡derivative

The derivative of a function represents the rate at which the function is changing at any given point. In the video, the concept of the derivative is mentioned in the context of verifying the correctness of the integral by differentiating the result and ensuring it matches the original function.

💡verification

Verification in calculus involves checking the result of a calculus operation, such as integration, by performing the reverse operation (differentiation) and ensuring the original function is obtained. The script suggests verifying the integral by differentiating the result to confirm that it equals 'e to the 3x plus 1'.

💡variable transformation

Variable transformation is a technique used in calculus to simplify integrals by changing the variable of integration. The video script uses variable transformation by substituting 'U' for '3x + 1', which simplifies the integral and demonstrates how to handle more complex integrals.

💡natural logarithm

The natural logarithm is the logarithm to the base 'e', where 'e' is an irrational number approximately equal to 2.71828. The video script mentions 'e' in the context of exponential functions, highlighting its importance in calculus as the base for natural logarithms and the exponential growth function.

Highlights

Introduction to evaluating the indefinite integral of e to the 3x plus 1.

Application of the antiderivative rule for e to the U.

Explanation of the constant of integration C.

Substitution of U as a function of X, where U equals 3x plus 1.

Derivation of D U as 3 times DX.

Adjustment for the absence of 3 DX in the integral by multiplying and dividing by 3.

Integration of e to the 3x plus 1 times 3 DX.

Substitution of the integral to 1/3 times the integral of e to the U.

Application of the integration formula to obtain one-third times e to the U plus C.

Conversion back to the original variable to get one-third e to the power of three X plus one plus C.

Emphasis on the importance of checking the derivative to ensure the correctness of the integral.

Instruction on taking the derivative of the answer to verify it matches the original function.

Conclusion of the process with a reminder to always check the derivative.