Integration by Parts with Natural Logarithms

The video tutorial explains how to evaluate the integral of the natural log of 2x with respect to x using integration by parts. It begins by discussing why basic integration formulas and substitution do not work for this problem. The tutorial then introduces the integration by parts formula, choosing u as the natural log function due to its difficulty in integration, and dv as the remaining part. The process involves differentiating u and integrating dv to find v, followed by applying the integration by parts formula to solve the integral. The tutorial concludes with the final solution and emphasizes the importance of choosing u and dv correctly.

What method is chosen to evaluate the integral of the natural log of 2x?

• Partial fraction decomposition

• Trigonometric substitution

• Integration by parts

• Integration by substitution

Why is the natural log function chosen as u in integration by parts?

• Because it is easy to differentiate

• Because it is easy to integrate

• Because it is a constant function

• Because it simplifies the integral

What is the derivative of the natural log of 2x?

• 1/(2x)

• 2/(2x)

• 1/x

• 2/x

What is the integral of dv when dv is equal to dx?

• ln(x)

• x

• 2x

• 1/x

What is the purpose of applying the chain rule in this integration?

• To find the constant of integration

• To integrate the natural log function

• To differentiate the natural log function

• To simplify the integral

What is the result of integrating x times 1/x?

• 0

• 1

• ln(x)

• x

What is the final antiderivative of the integral of the natural log of 2x?

• x ln(2x) + C

• x ln(2x) - x + C

• ln(2x) - x + C

• x ln(x) - x + C

What is the integral of 1 with respect to x?

• ln(x)

• 0

• 1

• x

Why is it important not to choose dv as the natural log function?

• Because it simplifies the integral

• Because it is a constant function

• Because it is difficult to integrate

• Because it is difficult to differentiate

What is the next step after finding the antiderivative in this example?

• Reviewing the integration by parts formula

• Looking at another example

• Finding the constant of integration

• Differentiating the result