Trigonometric Substitution in Integration

The video tutorial explains how to evaluate the indefinite integral of 1 over the square root of 1 minus x squared using trigonometric substitution. It begins by discussing why u substitution is ineffective and introduces trig substitution as a method to simplify the integral. The tutorial explains the substitution of x with sine of theta and the importance of domain constraints. It further defines theta in terms of arcsin and discusses the implications on the unit circle. The integral is simplified using trigonometric identities, leading to the final evaluation. The tutorial concludes by showing that the antiderivative is arcsin of x plus a constant.

What is the main reason for using trigonometric substitution in this integral problem?

• To change the variable of integration

• To make the integral more complex

• To simplify the integral using trigonometric identities

• To avoid using any substitution

Which trigonometric function is used to substitute x in this problem?

• Secant of theta

• Tangent of theta

• Sine of theta

• Cosine of theta

What is the domain constraint for x in this problem?

• x must be less than -1

• x can be any real number

• x must be greater than 1

• x must be between -1 and 1

Why is it important to consider the range of theta in this problem?

• To simplify the integral further

• To avoid complex numbers

• To ensure the substitution is valid

• To ensure theta is always positive

What does the arcsin function return in terms of angle range?

• Between 0 and pi

• Between -pi/2 and pi/2

• Between -pi and pi

• Between 0 and 2pi

What is the derivative of sine theta with respect to theta?

• Tangent theta

• Cosine theta

• Sine theta

• Secant theta

What is the principal root of cosine squared theta?

• Absolute value of cosine theta

• Negative cosine theta

• Cosine theta

• Sine theta

What does the integral of cosine theta over cosine theta simplify to?

• One

• Cosine theta

• Theta plus C

• Zero

What is the final result of the integral in terms of arcsin?

• Arcsin x plus C

• Arctan x plus C

• Arccos x plus C

• Arcsec x plus C

What is the significance of the result obtained from the integral?

• It confirms the integral is zero

• It proves the integral is undefined

• It shows the derivative of arcsin x

• It demonstrates a new trigonometric identity