Integration and Trigonometric Identities

This video tutorial explains how to find indefinite integrals using trigonometric substitution. It covers three main forms: square root of a squared minus x squared, a squared plus x squared, and x squared minus a squared. The video provides detailed examples of solving integrals using sine and tangent substitutions, including step-by-step algebraic simplifications and trigonometric identities. The tutorial also demonstrates how to convert back to the original variable using right triangle principles and inverse trigonometric functions.

Which substitution is used for the form sqrt(a^2 - x^2)?

• x = a tan(theta)

• x = a sin(theta)

• x = a sec(theta)

• x = a cos(theta)

In the example problem, what is the value of 'a' when integrating sqrt(4 - x^2) / x^2?

• 3

• 2

• 4

• 1

What is the reciprocal identity of 1/sin(theta)?

• sec(theta)

• cosec(theta)

• tan(theta)

• cot(theta)

What is the antiderivative of cosec^2(theta)?

• cot(theta)

• -cot(theta)

• -tan(theta)

• tan(theta)

How do you express theta in terms of x when x = 2 sin(theta)?

• theta = arcsec(x/2)

• theta = arctan(x/2)

• theta = arccos(x/2)

• theta = arcsin(x/2)

For the form x^2 + a^2, which substitution is appropriate?

• x = a cos(theta)

• x = a sin(theta)

• x = a tan(theta)

• x = a sec(theta)

What is the derivative of tan(theta) with respect to theta?

• tan^2(theta)

• cosec^2(theta)

• sec^2(theta)

• cot^2(theta)

In the second example, what is the value of 'a' when integrating x^3 / sqrt(x^2 + 9)?

• 4

• 3

• 2

• 1

What is the Pythagorean identity used to simplify 1 + tan^2(theta)?

• cosec^2(theta)

• tan^2(theta)

• sec^2(theta)

• cot^2(theta)

What is the final expression for sec(theta) in terms of x when x = 3 tan(theta)?

• sqrt(9 + x^2) / 3

• sqrt(9 + x^2) / x

• 3 / sqrt(9 + x^2)

• x / sqrt(9 + x^2)