Trigonometric Substitution in Integrals

The video tutorial explains integration using trigonometric substitution. It begins by discussing why basic U substitution is not applicable for the given integral. The tutorial then introduces trig substitution, leveraging Pythagorean identities to simplify the integration process. The instructor demonstrates the application of trig substitution to an integral involving x^2 + 9, using a reference triangle to aid in the process. The integral is simplified step-by-step, leading to the final anti-derivative expressed in terms of x. The tutorial concludes with a summary of the steps and the importance of the reference triangle in converting back to x.

Why is basic U substitution not suitable for the integral involving x^2 + 9?

• Because it requires partial fraction decomposition

• Because it involves trigonometric functions

• Because the differential does not match the integral form

• Because the integral is already simplified

Which trigonometric identity is used when the integral involves a^2 + x^2?

• x = a cos(θ)

• x = a tan(θ)

• x = a sin(θ)

• x = a sec(θ)

What substitution is made for x in the integral involving x^2 + 9?

• x = 3 sec(θ)

• x = 3 sin(θ)

• x = 3 cos(θ)

• x = 3 tan(θ)

In the reference triangle, what does the hypotenuse represent?

• x

• 3

• √(x^2 + 9)

• 9

What is the common factor that is factored out during the simplification of the integral?

• 9

• 3

• 2

• 5

What trigonometric identity is used to simplify the expression involving tangent squared plus one?

• sin^2(θ) + cos^2(θ) = 1

• tan^2(θ) + 1 = sec^2(θ)

• sec^2(θ) - tan^2(θ) = 1

• 1 + cot^2(θ) = csc^2(θ)

What substitution is made for U in the final integration step?

• U = sec(θ)

• U = tan(θ)

• U = cos(θ)

• U = sin(θ)

What is the final form of the anti-derivative in terms of x?

• 5x / √(x^2 + 9) + C

• -5√(x^2 + 9) / 9x + C

• 5√(x^2 + 9) / 9x + C

• -5x / √(x^2 + 9) + C

What is the role of the reference triangle in the final step?

• To simplify the integral

• To determine the limits of integration

• To find the derivative

• To convert the result back to x

What is the cosecant of θ in terms of x using the reference triangle?

• x / √(x^2 + 9)

• x / 3

• 3 / x

• √(x^2 + 9) / x