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

Summary

This 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.