Calculus II: Integration By Parts (Level 3 of 6)

The video tutorial covers integration by parts, focusing on problems requiring multiple applications of the formula. It provides a detailed solution for integrating x^2 * cos(x) and highlights potential errors in repeated applications. The tutorial introduces the tabular method as a shortcut for certain integrals, demonstrating its use with an example involving x^2 * sin(pi * x).

What is the first step in applying the integration by parts formula?

• Assign u and dv to the functions in the integrand

• Directly integrate the product of the functions

• Differentiate both functions

• Use the tabular method

What is the final result of integrating x squared times cosine of x using integration by parts?

• x squared times cosine of x plus 2x times sine of x minus 2 times cosine of x plus C1

• x squared times cosine of x minus 2x times sine of x plus 2 times cosine of x plus C1

• x squared times sine of x minus 2x times cosine of x plus 2 times sine of x plus C1

• x squared times sine of x plus 2x times cosine of x minus 2 times sine of x plus C1

Why is it important to maintain consistent substitutions in repeated applications of integration by parts?

• To avoid undoing previous integrations

• To ensure the integral converges

• To make the process faster

• To simplify the integrand

What is the tabular method best suited for?

• Integrals of the form x^n times sine, cosine, or exponential functions

• Integrals involving logarithmic functions

• Integrals involving trigonometric identities

• Integrals with rational functions

What is the first step in setting up the tabular method?

• Choose u and differentiate it

• Directly solve the integral

• Set up a table with u and its derivatives

• Choose dv and integrate it

In the tabular method, what is the purpose of alternating signs between rows?

• To match the number of derivatives

• To simplify the calculation

• To account for the alternating nature of derivatives and integrals

• To ensure the result is positive

What is the result of integrating x squared times sine of pi times x using the tabular method?

• x squared times 1 over pi times cosine of pi times x plus 2x times 1 over pi squared times sine of pi times x minus 2 times 1 over pi cubed times cosine of pi times x plus C

• Negative x squared times 1 over pi times sine of pi times x plus 2x times 1 over pi squared times cosine of pi times x minus 2 times 1 over pi cubed times sine of pi times x plus C

• x squared times 1 over pi times sine of pi times x minus 2x times 1 over pi squared times cosine of pi times x plus 2 times 1 over pi cubed times sine of pi times x plus C

• Negative x squared times 1 over pi times cosine of pi times x plus 2x times 1 over pi squared times sine of pi times x plus 2 times 1 over pi cubed times cosine of pi times x plus C