Integration by Parts Concepts

The video tutorial explains how to find the indefinite integral of x cosine squared x times tangent x. It begins by simplifying the expression using trigonometric identities, then applies the integration by parts method. The tutorial provides a step-by-step guide, including the use of the double angle formula and integration by parts formula, to arrive at the final solution. The process is broken down into manageable steps, making it easier to understand the application of calculus techniques in solving integrals.

What is the first step in simplifying the integral of x cosine squared x times tangent x?

• Use the Pythagorean identity

• Differentiate the expression

• Apply the power rule

• Cancel out a cosine term

How is the double angle formula used in this problem?

• To simplify cosine squared x

• To convert tangent x into sine x over cosine x

• To find the derivative of sine 2x

• To express sine x cosine x as sine 2x

What is the purpose of using integration by parts in this problem?

• To differentiate the expression

• To simplify the expression further

• To apply the chain rule

• To find the antiderivative of a product of functions

Which formula is used to find the integral of u dv?

• u minus v plus the integral of v du

• u times v plus the integral of v du

• u plus v minus the integral of v du

• u times v minus the integral of v du

Which variable is chosen as 'u' in the integration by parts process?

• sine 2x

• dx

• cosine 2x

• x

What is the antiderivative of sine 2x?

• negative sine 2x divided by 2

• sine 2x divided by 2

• negative cosine 2x divided by 2

• cosine 2x divided by 2

What is the result of distributing the one-half in the integration by parts calculation?

• One-half x sine 2x plus one-half integral of sine 2x dx

• Negative one-half x cosine 2x plus one-half integral of cosine 2x dx

• One-fourth x sine 2x plus one-fourth integral of sine 2x dx

• Negative one-fourth x cosine 2x plus one-fourth integral of cosine 2x dx

What is the derivative of 2x used in the integration by parts?

• x

• 2

• 0

• 1

What is the final antiderivative of the integral of x cosine squared x times tangent x?

• 1/4 x sine 2x plus 1/8 cosine 2x plus C

• Negative 1/4 x sine 2x plus 1/8 cosine 2x plus C

• Negative 1/4 x cosine 2x plus 1/8 sine 2x plus C

• 1/4 x cosine 2x plus 1/8 sine 2x plus C

What is the constant of integration in the final answer?

• C

• 0

• 2

• 1