Integration Techniques and Applications

The video tutorial explains how to integrate the arc sine of x using integration by parts. It begins by setting up the integration by parts formula, identifying u and dv, and calculating the derivative of arc sine. The tutorial then applies the integration by parts formula and uses u-substitution to simplify the integral. Finally, it completes the integration process and presents the final solution.

What is the first step in solving the integral of arc sine of x?

• Apply the power rule

• Use integration by parts

• Differentiate the function

• Use the substitution method

In the integration by parts formula, what is chosen as u?

• x

• arc sine of x

• dx

• 1 minus x squared

What is the derivative of arc sine of x?

• x over the square root of 1 minus x squared

• 1 over the square root of 1 minus x squared

• 1 over x

• x squared over 1 minus x

What substitution is used to simplify the integral after applying integration by parts?

• u = x squared

• u = arc sine of x

• u = x

• u = 1 minus x squared

How is the integral of 1 over the square root of u expressed using the power rule?

• u to the power of -1/2

• u to the power of 1/2

• u to the power of 1

• u to the power of 2

What is the purpose of dividing by negative 2x during the u-substitution process?

• To simplify the expression

• To eliminate x from the equation

• To change the sign of the integral

• To make the integral positive

What happens to the x terms during the simplification of the integral?

• They are canceled out

• They are multiplied

• They are added together

• They are divided

What is the final form of the integral of arc sine of x?

• x squared plus arc sine of x plus c

• arc sine of x plus x squared plus c

• x arc sine of x minus the square root of 1 minus x squared plus c

• x arc sine of x plus the square root of 1 minus x squared plus c

What is the result of applying the power rule to u to the negative one-half?

• u to the power of 1

• u to the power of -1

• u to the power of 0

• u to the power of 1/2

What constant is added to the final integral result?

• c

• 0

• 1

• x