Definite Integrals and Integration Techniques

This video tutorial explains the process of solving a definite integral using integration by parts. It begins with an introduction to the integration by parts formula and how to identify the components U and DV in the integrand. The tutorial then demonstrates finding the indefinite integral and using it to evaluate the definite integral. The video concludes with the final calculations and a summary of the example.

What is the formula for integration by parts?

• Integral of U DV = UV - Integral of V DU

• Integral of U DV = U*V + Integral of V DU

• Integral of U DV = UV + Integral of V DU

• Integral of U DV = U/V - Integral of V DU

When identifying U in integration by parts, which function is typically chosen?

• The function with the highest degree

• The function with the lowest degree

• The function that is easiest to differentiate

• The function that is easiest to integrate

What is the next step after identifying U and DV in integration by parts?

• Integrate U and differentiate DV

• Integrate both U and DV

• Differentiate U and integrate DV

• Differentiate both U and DV

What is the indefinite integral of 2x^2 with respect to x?

• 1/3 x^3

• 2x^3

• x^3

• 2/3 x^3

How do you evaluate a definite integral using an indefinite integral?

• By integrating the indefinite integral again

• By differentiating the indefinite integral

• By multiplying the indefinite integral by the limits

• By substituting the limits into the indefinite integral

What happens to the constant of integration when evaluating a definite integral?

• It is subtracted from the result

• It is multiplied by the limits

• It is added to the result

• It cancels out

What is the value of the natural log of 1?

• 1

• 0

• e

• Infinity

What is the common denominator used to simplify the terms in the final result?

• 9

• 12

• 6

• 3

What is the exact value of the definite integral from 1 to e of 2x ln(x) dx?

• 2/9 e^3 + 4/9

• 4/9 e^3 + 2/9

• 4/9 e^3 - 2/9

• 2/9 e^3 - 4/9

What is the approximate decimal value of the definite integral evaluated?

• 7.14913

• 9.14913

• 8.14913

• 10.14913