Integration of Sine to the Fifth Power

The video tutorial explains how to integrate sine to the fifth power of X using u-substitution. It begins by splitting sine to the fifth power into sine to the fourth times sine, then applies u-substitution by setting u equal to cosine X. The integral is simplified by canceling terms and using algebraic techniques. The antiderivative is found for each term in the polynomial, and the final expression is written in standard form, resulting in the integral of sine raised to the fifth power.

Why do we split sine to the fifth power into two parts?

• To simplify the expression

• To make it easier to differentiate

• To apply the chain rule

• To facilitate u-substitution

What identity is used to replace sine squared?

• Sine squared equals 1 minus cosine squared

• Sine squared equals cosine squared

• Sine squared equals tangent squared

• Sine squared plus cosine squared equals 1

What is the purpose of using u-substitution in this integration?

• To simplify the integral into a polynomial

• To find the derivative

• To change the variable of integration

• To eliminate the trigonometric function

What happens to the sine term during the simplification process?

• It is integrated directly

• It is squared

• It is differentiated

• It is canceled out

What method is used to expand the integral?

• Chain rule

• Product rule

• Integration by parts

• FOIL method

What is the result of combining like terms in the expanded integral?

• A constant

• A polynomial expression

• A trigonometric identity

• A single term

What is the antiderivative of u squared?

• U to the fourth over 4

• U squared over 2

• U cubed over 3

• U to the fifth over 5

What is the final step after finding the antiderivative?

• Apply the chain rule

• Substitute back the original variable

• Differentiate the result

• Simplify the expression

What is the coefficient of cosine to the fifth power in the final answer?

• 1/5

• -1/5

• -2/3

• 2/3

What constant is added to the final antiderivative?

• C

• D

• F

• E