Integration of Sine Natural Logarithm

This video tutorial explains how to find the integral of sine natural log of X using substitution and integration by parts. The process involves substituting U for Ln X, simplifying the expression, and applying integration by parts twice to reach the final solution. The tutorial concludes with the derivation of the final answer, demonstrating the step-by-step approach to solving complex integrals.

What is the first step in solving the integral of sine natural log of X?

• Use partial fraction decomposition

• Apply use substitution

• Differentiate the function

• Use trigonometric identities

In the substitution method, what is U set equal to?

• Ln X

• sin X

• e^X

• X

What technique is used after substitution to solve the integral?

• Partial fraction decomposition

• Integration by parts

• Trigonometric substitution

• Differentiation

In integration by parts, what is the formula used?

• U/V

• U + V

• UV - integral of VDU

• U * V

What is the integral of sine in the context of integration by parts?

• Cosine

• Negative cosine

• Tangent

• Negative tangent

What happens to the two negative signs in the expression during simplification?

• They add up

• They cancel each other

• They multiply

• They remain unchanged

What is the purpose of multiplying both sides by 1/2 in the final steps?

• To simplify the expression

• To balance the equation

• To factor out a common term

• To convert to a different base

What is the final expression of the integral of sine natural log of X?

• X times sine Ln X minus cosine Ln X

• 1/2 X times sine Ln X minus cosine Ln X

• X times sine Ln X plus cosine Ln X

• 1/2 X times sine Ln X plus cosine Ln X

What is the role of e to the Ln X in the final expression?

• It is factored out

• It becomes zero

• It remains unchanged

• It simplifies to X

What constant is added to the final expression of the integral?

• B

• A

• C

• D