Understanding Anti-Derivatives and Substitution

The video tutorial explains how to find the anti-derivative of a function involving arcsin(x). It begins by recognizing the derivative of arcsin(x) and then uses substitution to simplify the integral. The process involves changing variables to U, integrating, and then substituting back to express the result in terms of x. The final anti-derivative is presented, demonstrating the relationship between the derivative and the original function.

What is the initial function for which we are trying to find the anti-derivative?

• R cos x raised to the power of 3 divided by the square root of 1 - x^2

• R sin x raised to the power of 3 divided by the square root of 1 - x^2

• R cos x raised to the power of 2 divided by the square root of 1 - x^2

• R sin x raised to the power of 2 divided by the square root of 1 - x^2

What is the derivative of R sin x?

• 1 / the square root of x^2 - 1

• 1 / the square root of x^2 + 1

• 1 / the square root of 1 - x^2

• 1 / the square root of 1 + x^2

What substitution is made to simplify the integral?

• Let U equal R cos x

• Let U equal R sin x

• Let U equal R tan x

• Let U equal R cot x

After substitution, what does the integral become in terms of U?

• Integral of U to the fourth du

• Integral of U squared du

• Integral of U to the fifth du

• Integral of U cubed du

What is the anti-derivative of U squared in terms of U?

• U to the fifth divided by 5 plus C

• U to the fourth divided by 4 plus C

• U squared divided by 2 plus C

• U cubed divided by 3 plus C

What is the final anti-derivative in terms of x?

• R sin x squared divided by 2 plus C

• R sin x to the fifth divided by 5 plus C

• R sin x cubed divided by 3 plus C

• R sin x to the fourth divided by 4 plus C

What does the anti-derivative function represent?

• The derivative of this function is equal to zero

• The derivative of this function is equal to infinity

• The derivative of this function is equal to a constant

• The derivative of this function is equal to the original integrand

What is the purpose of finding the anti-derivative in terms of x?

• To express the solution in the original variable

• To simplify the equation

• To eliminate the constant of integration

• To find the maximum value of the function