Integrating Trigonometric Functions

This video tutorial demonstrates integration using trigonometric substitution. It begins by explaining why basic U substitution is not applicable and introduces trigonometric substitution as a solution. The process involves setting up a reference triangle and performing the substitution, followed by simplifying the integral using trigonometric identities. The tutorial concludes with the final integration steps and converting the result back to the original variable.

Why is basic U substitution not suitable for the given integral?

• The integral contains exponential functions.

• U substitution is only for polynomial integrals.

• The integral is already in its simplest form.

• The derivative of U does not match the integral's form.

What substitution is used instead of x = a secant Theta?

• x = a tangent Theta

• x = a sine Theta

• 2x = a secant Theta

• 2x = a cosine Theta

What is the purpose of drawing a reference triangle in trigonometric substitution?

• To determine the relationships between the sides and angles

• To visualize the trigonometric identities

• To find the limits of integration

• To simplify the integral directly

How is the integral simplified after substitution?

• By using trigonometric identities

• By expanding the polynomial

• By differentiating the expression

• By applying the chain rule

What is the result of simplifying the square root in the integral?

• 5 sine Theta

• 4 cosine Theta

• 2 secant Theta

• 3 tangent Theta

Which formula is applied to integrate sin^2 Theta?

• Sum-to-product formula

• Double angle formula

• Product-to-sum formula

• Power-reducing formula

What substitution is used to integrate cosine 2 Theta?

• U = 2 Theta

• U = Theta/2

• U = 3 Theta

• U = Theta^2

How is Theta expressed in terms of x for the final result?

• Theta = arc sine (3/2x)

• Theta = arc cosine (3/2x)

• Theta = arc tangent (3/2x)

• Theta = arc secant (3/2x)

What is the expression for sin Theta using the reference triangle?

• 2x / 3

• Square root of (9 - 4x^2) / 3

• 3 / 2x

• Square root of (4x^2 - 9) / 2x

What is the final form of the anti-derivative?

• 2/3 arc cosine (3/2x) - square root of (4x^2 - 9) / 2x^2 + C

• 2/3 arc sine (3/2x) - square root of (4x^2 - 9) / 2x^2 + C

• 2/3 arc secant (3/2x) - square root of (4x^2 - 9) / 2x^2 + C

• 2/3 arc tangent (3/2x) - square root of (4x^2 - 9) / 2x^2 + C