Eric's Calculus Lecture: Evaluate the Indefinite Integral ∫e^(3x+1)dx
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
📚 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
💡antiderivative
💡integration rule
💡constant of integration
💡u-substitution
💡Du
💡exponential function
💡derivative
💡verification
💡variable transformation
💡natural logarithm
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.
Transcripts
okay let's evaluate the indefinite
integral of e to the 3x plus 1 now the
integration rule that we're going to
apply is a very very easy one it's the
antiderivative or the indefinite
integral of e to the U du you because
equal to e to the U plus C don't forget
the constant of integration ok so
remember you but we write U but U is
really U as a function of X so the U is
like it's like at f of X all right so
let's let's apply a quick u substitution
here and in our integral let's say U is
equal to 3x plus 1 which means D u
should be equal to 3 times DX well we
don't have a 3 DX in our integral we
have only DX so let's do a let's do a
little multiplying and dividing and
that's before we eat before we make our
substitution let's say let's bring it
down here this is the integral of e to
the 3x plus 1 times 3 DX alright perfect
so now we have our you but we can't just
multiply by 3 without bouncing off that
change with the 1/3 okay let's do
something like that ok so now we have
our D you and we have our you okay so
upon making the substitution now we are
at 1/3 times the integral
e to the u d-- u which is precisely
one-third times e so they you applying
this formula up here and plus c don't
forget the constant and let's go back to
our original variable so this is
one-third e to the power three X plus
one plus C and you can always check this
by taking the derivative taking the
derivative of your answer and making
sure you get back exactly what you
started with so if you take the
derivative of this down here you should
get exactly e so the 3x plus one okay
that about does it for this one
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