Trigonometric Substitution and Integrals

This video tutorial demonstrates integration using trigonometric substitution. It begins by explaining why basic U substitution is not applicable and proceeds to convert the expression into radical form. The tutorial then introduces trigonometric substitution, sketches a reference triangle, and performs the substitution. The integral is simplified using trigonometric identities, and the final integration is completed by converting back to the variable x. The tutorial concludes with the antiderivative expressed in terms of x.

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

• Because the derivative does not match the integral's form.

• Because it leads to an undefined expression.

• Because it results in a complex number.

• Because it requires partial fraction decomposition.

What is the purpose of rewriting the integral in radical form?

• To simplify the expression for easier integration.

• To eliminate the square root.

• To identify the correct trigonometric substitution.

• To convert it into a polynomial.

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

• x = a sec(θ)

• x = a cos(θ)

• x = a tan(θ)

• x = a sin(θ)

What is the expression for x in terms of θ after substitution?

• x = 3/4 sin(θ)

• x = 3 sin(θ)

• x = 4/3 sin(θ)

• x = 4 sin(θ)

What is the role of the reference triangle in trigonometric substitution?

• To determine the limits of integration.

• To verify the solution of the integral.

• To find the relationship between x and θ.

• To calculate the derivative of the function.

After substitution, what does the integral simplify to before final integration?

• 4/3 * integral of sin(θ) dθ

• 4/3 * integral of tan(θ) dθ

• 4/3 * integral of cos(θ) dθ

• 4/3 * integral of sec(θ) dθ

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

• tan(θ)

• cos(θ)

• sin(θ)

• cot(θ)

How is the final result expressed in terms of x?

• By differentiating the result with respect to x.

• Using the original substitution equation.

• By using the reference triangle to find tangent(θ).

• By integrating again with respect to x.

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

• 3x / (16 * sqrt(16 - 9x^2)) + C

• x / (16 * sqrt(16 - 9x^2)) + C

• x / (4 * sqrt(16 - 9x^2)) + C

• 3x / (4 * sqrt(16 - 9x^2)) + C

What is the constant of integration denoted by?

• K

• C

• B

• D