Integration Problems with Trig Substitution

This video tutorial covers advanced integration problems using trigonometric substitution. It reviews the three main types of integrals that require trig substitution and provides two detailed examples. The first example involves completing the square to fit a given form, while the second example uses trigonometric identities to simplify the integration process. The video concludes with a summary of the techniques discussed.

Which of the following is a type of integral that typically requires trig substitution?

• Integral of ln(x)

• Integral of e^x

• Integral of x^2

• Integral involving the square root of a^2 - x^2

What is the first step in completing the square for the expression x^2 + 6x?

• Multiply by 2

• Add 6 to both sides

• Take half of 6 and square it

• Subtract 9 from both sides

In the trig substitution for the integral involving x^2 + 6x, what substitution is made for x + 3?

• 3 sec(Theta)

• 3 tan(Theta)

• 3 sin(Theta)

• 3 cos(Theta)

What is the purpose of using a right triangle in the context of trig substitution?

• To calculate the area

• To model the angle Theta

• To find the hypotenuse

• To determine the perimeter

What identity is used when performing trig substitution for the integral involving 16 - x^2?

• sin^2(Theta) + cos^2(Theta) = 1

• tan^2(Theta) + 1 = sec^2(Theta)

• cos^2(Theta) - sin^2(Theta) = 1

• 1 - sin^2(Theta) = cos^2(Theta)

What substitution is made for x in the integral involving 16 - x^2?

• x = 4 cos(Theta)

• x = 4 tan(Theta)

• x = 4 sin(Theta)

• x = 4 sec(Theta)

What is the result of simplifying the integral after performing the trig substitution for 16 - x^2?

• Integral of cos^2(Theta)

• Integral of sin^2(Theta)

• Integral of tan^2(Theta)

• Integral of sec^2(Theta)

What is the differential dx equal to in the substitution x = 4 sin(Theta)?

• 4 tan(Theta) dTheta

• 4 sec(Theta) dTheta

• 4 cos(Theta) dTheta

• 4 sin(Theta) dTheta

Which trigonometric identity is used to simplify the integral of sin^2(Theta)?

• sin^2(Theta) = 1 - cos(2Theta)/2

• cos^2(Theta) = 1 + sin(2Theta)/2

• tan^2(Theta) = sec^2(Theta) - 1

• sec^2(Theta) = 1 + tan^2(Theta)

What is the final expression for the integral in terms of x after using trig substitution?

• 8 * inverse sin(x/4) - x * sqrt(16 - x^2) / 2 + C

• 8 * inverse tan(x/4) - x * sqrt(16 - x^2) / 2 + C

• 8 * inverse cos(x/4) + x * sqrt(16 - x^2) / 2 + C

• 8 * inverse sec(x/4) + x * sqrt(16 - x^2) / 2 + C