Integral of Sine Cube X

This lesson covers finding the integral of sine cubed X. It begins by expanding the expression using sine squared and the Pythagorean identity. The lesson then applies substitution, setting u as cosine X, to simplify the integral. The antiderivative is calculated, and the final solution is obtained by replacing variables back to cosine X.

What is the first step in finding the integral of sine cube X?

• Use the derivative of sine

• Use the double angle formula

• Directly integrate sine cube

• Expand sine cube as sine squared times sine

Which identity is used to express sine squared in terms of cosine?

• Half angle identity

• Sum of angles identity

• Pythagorean identity

• Double angle identity

What substitution is made to simplify the integral?

• u = secant X

• u = sine X

• u = tangent X

• u = cosine X

What is the antiderivative of 1 with respect to u?

• u^2/2

• ln(u)

• u

• 1/u

How is the negative sign handled in the integration process?

• Ignored

• Moved to the front of the integral

• Multiplied with the integrand

• Added to the constant of integration

What is the antiderivative of u squared?

• u^3/3

• u^2/2

• 3u^2

• 2u

What is the final expression for the integral of sine cube X?

• 1/3 cosine cube X - cosine X + C

• cosine cube X - 1/3 cosine X + C

• 1/3 sine cube X - sine X + C

• sine cube X - 1/3 sine X + C

What is the role of the constant C in the final answer?

• It represents a specific solution

• It accounts for the constant of integration

• It is a placeholder for zero

• It is used to balance the equation

Why is u replaced with cosine X in the final step?

• To simplify the expression

• To revert back to the original variable

• To make the expression more complex

• To eliminate the constant C