Trigonometric Substitution and Integration

This video tutorial covers integration using trigonometric substitution, focusing on cases where completing the square is necessary. It begins by explaining why basic u substitution is not applicable, then details the process of completing the square to fit the form required for trig substitution. The tutorial includes setting up the substitution using a reference triangle, simplifying the integral with trigonometric identities, and performing back substitution to express the solution in terms of x. The video concludes with a discussion on simplifying the final answer.

Why is basic u-substitution not suitable for the given integral?

• The integrand is already simplified.

• The integrand is a polynomial.

• The integrand does not contain a constant term.

• The integrand contains a constant term.

Which trigonometric substitution is used when the square root is in the form of a^2 - x^2?

• x = a sec(θ)

• x = a sin(θ)

• x = a cos(θ)

• x = a tan(θ)

What is the purpose of completing the square in the context of this problem?

• To change the variable of integration.

• To fit the expression into a known trigonometric form.

• To eliminate the square root.

• To simplify the integrand.

After completing the square, what substitution is made for x?

• x = 2 sin(θ) + 3

• x = 2 sec(θ) + 3

• x = 2 tan(θ) + 3

• x = 2 cos(θ) + 3

What is the simplified form of the square root after substitution?

• 2 tan(θ)

• 2 cos(θ)

• 2 sec(θ)

• 2 sin(θ)

Which trigonometric identity is used to simplify the expression under the square root?

• tan^2(θ) + 1 = sec^2(θ)

• sec^2(θ) - tan^2(θ) = 1

• sin^2(θ) + cos^2(θ) = 1

• 1 + cot^2(θ) = csc^2(θ)

What is the integral of 2 sec^2(θ) with respect to θ?

• 2 tan(θ)

• 2 cos(θ)

• 2 sec(θ)

• 2 sin(θ)

How is sec(θ) expressed in terms of x after substitution?

• 2x - 3

• 2 / (x - 3)

• (x - 3) / 2

• x / 2

What is the final form of the antiderivative in terms of x?

• 2 tan(θ) + 3 ln|x - 3| + C

• 2 tan(θ) + 3 ln|sec(θ) + tan(θ)| + C

• 2 tan(θ) + 3 ln|x - 3 + sqrt(x^2 - 6x + 5)| + C

• 2 tan(θ) + 3 ln|x + 3| + C

Why can the constant 3 ln(1/2) be absorbed into the constant of integration?

• It simplifies the expression.

• It is a constant value.

• It is part of the original function.

• It is a variable term.