Inverse Sine Integrals and Applications

The video tutorial explains how to evaluate a definite integral involving an inverse sine function. It begins by analyzing the integrand and considering the use of a specific integration formula. However, due to the presence of an inverse sine in the numerator, a substitution method is applied. The tutorial details the substitution process, including finding the derivative and adjusting the limits of integration. Finally, the integral is evaluated, and the exact value is calculated, demonstrating the complete solution to the problem.

What is the initial form of the integrand function in the given problem?

• u squared divided by the square root of a squared minus u squared

• x squared divided by the square root of one minus x squared

• u divided by the square root of a squared minus u squared

• Inverse sine of x divided by the square root of one minus x squared

Which formula should remind us of the derivative of inverse sine?

• The derivative of cosine u with respect to x

• The integral of inverse sine x

• The derivative of inverse sine u with respect to x

• The integral of u squared

What substitution is made for the variable u in the integral?

• u equals x squared

• u equals inverse sine of x

• u equals cosine of x

• u equals the square root of x

What is the coefficient of inverse sine x in the integrand function after substitution?

• Zero

• Two

• One

• Pi

What is the lower limit of integration with respect to u when x is zero?

• Zero

• Pi over six

• Pi over two

• One

What angle corresponds to the upper limit of integration when x is one half?

• 30 degrees

• 90 degrees

• 45 degrees

• 60 degrees

In which quadrant is the angle for the upper limit of integration located?

• Fourth quadrant

• Second quadrant

• First quadrant

• Third quadrant

What is the significance of the 30-60-90 triangle in this context?

• It helps determine the lower limit of integration

• It helps determine the upper limit of integration

• It is used to calculate the derivative

• It is used to find the anti-derivative

What is the anti-derivative of u with respect to u?

• u cubed divided by three

• u squared divided by two

• u squared

• u divided by two

What is the final value of the definite integral?

• Pi divided by 6

• Pi squared divided by 12

• Pi squared divided by 72

• Pi squared divided by 36