Trigonometric Integration Techniques

This video is the third in a series on integration using trigonometric substitution. It begins with a review of previous methods for integrating functions involving square roots of the form a^2 - x^2 and a^2 + x^2. The focus of this video is on integrals involving the square root of x^2 - a^2, using the substitution x = a sec(theta). The video walks through an example problem, demonstrating the setup, substitution, simplification, and integration process. The solution is then expressed in terms of x, and the video concludes with a preview of more challenging integrals to be covered in the next part.

What substitution is used for integrals involving the square root of a^2 - x^2?

• x = a tan(θ)

• x = a sin(θ)

• x = a cos(θ)

• x = a sec(θ)

Which identity is used when substituting x = a sec(θ) for integrals involving the square root of x^2 - a^2?

• sec^2(θ) - 1 = tan^2(θ)

• tan^2(θ) + 1 = sec^2(θ)

• sin^2(θ) + cos^2(θ) = 1

• cos^2(θ) - sin^2(θ) = 1

Before using trigonometric substitution, what should you check for?

• Differentiation

• Integration by parts

• Partial fraction decomposition

• Basic u-substitution

In the example problem, what is the value of 'a' when dealing with the square root of x^2 - 25?

• 6

• 5

• 4

• 3

What is the differential dx when x = 5 sec(θ)?

• 5 sec(θ) tan(θ) dθ

• 5 cos(θ) dθ

• 5 tan(θ) dθ

• 5 sec(θ) dθ

How is tangent squared theta replaced in the integral?

• With 1 - sec^2(θ)

• With tan^2(θ) + 1

• With sin^2(θ) + cos^2(θ)

• With sec^2(θ) - 1

What is the antiderivative of sec^2(θ)?

• cos(θ)

• sec(θ)

• sin(θ)

• tan(θ)

How do you express theta in terms of x after integration?

• theta = cos^(-1)(x/5)

• theta = sec^(-1)(x/5)

• theta = tan^(-1)(x/5)

• theta = sin^(-1)(x/5)

What is the final expression for the integral in terms of x?

• sqrt(x^2 - 25) + 5 tan^(-1)(x/5) + C

• sqrt(x^2 - 25) - 5 sec^(-1)(x/5) + C

• sqrt(x^2 - 25) - 5 tan^(-1)(x/5) + C

• sqrt(x^2 - 25) + 5 sec^(-1)(x/5) + C

What will the next video cover?

• Differentiation techniques

• Applications of integration

• More challenging integrals requiring completing the square

• Basic integration techniques