Understanding Trigonometric Substitution in Integration

Understanding Trigonometric Substitution in Integration

Assessment

Interactive Video

Mathematics

11th Grade - University

Practice Problem

Hard

CCSS
HSF-BF.B.4D

Standards-aligned

Created by

Sophia Harris

FREE Resource

Standards-aligned

CCSS.HSF-BF.B.4D
The video tutorial explains how to evaluate the indefinite integral of 1/(1-x^2) using trigonometric substitution. It begins by discussing why traditional substitution methods may not work and introduces trig substitution as an alternative. The tutorial covers the necessary domain and range considerations, demonstrates the substitution process, and simplifies the expression using trigonometric identities. Finally, it completes the integration and provides insights into the derivative of the arcsine function.

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10 questions

Show all answers

1.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What is the initial problem discussed in the video?

Evaluating the indefinite integral of 1/(1-x^2)

Evaluating the definite integral of 1/(1-x^2)

Finding the derivative of 1/(1-x^2)

Solving the equation 1/(1-x^2) = 0

2.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Which substitution method is initially considered but found unsuitable?

Integration by parts

Partial fraction decomposition

Trigonometric substitution

U-substitution

3.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What trigonometric identity is used for substitution?

sin(theta) = cos(theta)

1 + cot^2(theta) = csc^2(theta)

tan^2(theta) + 1 = sec^2(theta)

sin^2(theta) + cos^2(theta) = 1

4.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What substitution is made for x in terms of theta?

x = tan(theta)

x = sec(theta)

x = sin(theta)

x = cos(theta)

Tags

CCSS.HSF-BF.B.4D

5.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What is the range of theta when using arcsin(x) for substitution?

0 to 2pi

-pi to pi

-pi/2 to pi/2

0 to pi

Tags

CCSS.HSF-BF.B.4D

6.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Why is the domain of x restricted to between -1 and 1?

To match the range of arcsin

To simplify the calculation

To ensure the integral converges

To avoid division by zero

7.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What is the expression for dx in terms of theta?

dx = sin(theta) d(theta)

dx = sec(theta) d(theta)

dx = cos(theta) d(theta)

dx = tan(theta) d(theta)

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