Integration Techniques and Concepts

The video tutorial explains how to integrate the function 2x sine x using the integration by parts method. 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 and provides guidelines for selecting the parts of the integral to assign to u and dv. The process of applying the formula is demonstrated, leading to the simplification of the antiderivative. The video concludes with a brief mention of using integration by parts twice in a future example.

What is the main reason for choosing integration by parts for the integral of 2x sin(x)?

• It fits a basic integration formula.

• Substitution method works well.

• Integration by parts is the only method that works.

• It is the simplest method available.

In the integration by parts formula, what do u and dv represent?

• u is the function and dv is the limit.

• u is the constant and dv is the variable.

• u is a part of the integral and dv is the other part.

• u is the integral and dv is the derivative.

Why is u chosen as 2x in this example?

• Because its differential is more complex.

• Because its differential is simpler.

• Because it is easier to integrate.

• Because it is a constant.

What is the integral of sin(x) with respect to x?

• -sin(x)

• -cos(x)

• cos(x)

• sin(x)

What is the result of applying the integration by parts formula to the integral of 2x sin(x)?

• -2x sin(x) + 2 cos(x) + C

• 2x cos(x) - 2 sin(x) + C

• 2x sin(x) - 2 cos(x) + C

• -2x cos(x) + 2 sin(x) + C

What is the purpose of factoring out a 2 in the final expression?

• To simplify the expression.

• To eliminate the constant.

• To make the expression more complex.

• To change the integral.

What is the antiderivative of cos(x)?

• -sin(x)

• sin(x)

• cos(x)

• -cos(x)

What is the final form of the antiderivative after factoring?

• 2(sin(x) - x cos(x)) + C

• 2(cos(x) - x sin(x)) + C

• 2(x sin(x) - cos(x)) + C

• 2(x cos(x) - sin(x)) + C

What is mentioned as a future topic at the end of the video?

• Using substitution method.

• Using integration by parts twice.

• Using numerical integration.

• Using basic integration formulas.

What is the constant of integration denoted by?

• B

• D

• A

• C