Trigonometric Substitution in Integration

This video tutorial explains the process of integration using trigonometric substitution. It begins by discussing why basic u-substitution is not applicable and introduces trig substitution as an alternative. The tutorial sets up the substitution using a reference triangle and simplifies the integral using trigonometric identities. It further applies a power-reducing formula to simplify the integral before performing the final integration and substitution back to the original variable. The tutorial concludes with the derivation of the antiderivative.

Why is basic substitution not suitable for the given integral?

• The integral is too complex.

• The integral involves a trigonometric function.

• The differential does not match the numerator.

• The integral is already simplified.

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

• x = a tan(θ)

• x = a cos(θ)

• x = a sec(θ)

• x = a sin(θ)

In the reference triangle, what does the hypotenuse represent when x = 5 sin(θ)?

• x

• 5

• √(25 - x^2)

• θ

What identity is used to simplify the expression involving sin^2(θ)?

• sin^2(x) = 1 + cos(2x)

• sin^2(x) = cos^2(x)

• sin^2(x) = 1/2 * (1 - cos(2x))

• sin^2(x) = 1 - cos^2(x)

What is the result of integrating 1 with respect to θ?

• θ

• tan(θ)

• cos(θ)

• sin(θ)

What substitution is used to integrate cos(2θ)?

• u = 2θ

• u = θ

• u = sin(θ)

• u = cos(θ)

Which inverse trigonometric function is used to express θ in terms of x?

• arcsin(x/5)

• arcsec(x/5)

• arctan(x/5)

• arccos(x/5)

What is the expression for cos(θ) in terms of x?

• √(x^2 - 25)/5

• 5/x

• √(25 - x^2)/5

• x/5

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

• 25 arcsin(x/5) - x√(25 - x^2) + C

• 25 arcsec(x/5) + x√(25 - x^2) + C

• 25 arccos(x/5) + x√(25 - x^2) + C

• 25 arctan(x/5) - x√(25 - x^2) + C

What is the purpose of using trigonometric substitution in integration?

• To directly find the derivative.

• To simplify the integral by converting it into a trigonometric form.

• To make the integral more complex.

• To avoid using any trigonometric identities.