For the indefinite integral below, choose which of the following substitutions would be most
helpful in evaluating the integral.
Enter the appropriate letter (A,B, or C) in each blank.
A. x= 3tan(theta)
B. x= 3sin(theta)
C. x= 3sec(theta)
integrate of (dx)/((9-x^2)^3/2)
Solution
Selecting the option B. [x = 3sin theta =>] [dx = 3 cos theta d theta] yields:
[int (3 cos theta d theta)/((9-9sin^2 theta)sqrt(9 - 9 sin theta))]
Factoring out 9 yields:
[int (3 cos theta d theta)/(27(1-sin^2 theta)sqrt(1 - sin^2 theta))]
Using the fundamental formula of trigonometry yields:
[1-sin^2 theta = cos^2 theta]
[int (3 cos theta d theta)/(27cos^3 theta) = (1/9) int(d theta)/(cos^2 theta)]
[(1/9) int (d theta)/(cos^2 theta) = (1/9) tan theta + c]
Substituting back [(sin^(-1)(x/3))] for [theta] yields:
[int (dx)/((9-x^2)^3/2)= (1/9) tan((sin^(-1)(x/3)))+ c]
Hence, evaluating the given integral yields [int (dx)/((9-x^2)^3/2)= (1/9) tan((sin^(-1)(x/3))) + c]
, using the option B. [x = 3sin theta] ..
For the indefinite integral below, choose which of the following sub.pdf
1. For the indefinite integral below, choose which of the following substitutions would be most
helpful in evaluating the integral.
Enter the appropriate letter (A,B, or C) in each blank.
A. x= 3tan(theta)
B. x= 3sin(theta)
C. x= 3sec(theta)
integrate of (dx)/((9-x^2)^3/2)
Solution
Selecting the option B. [x = 3sin theta =>] [dx = 3 cos theta d theta] yields:
[int (3 cos theta d theta)/((9-9sin^2 theta)sqrt(9 - 9 sin theta))]
Factoring out 9 yields:
[int (3 cos theta d theta)/(27(1-sin^2 theta)sqrt(1 - sin^2 theta))]
Using the fundamental formula of trigonometry yields:
[1-sin^2 theta = cos^2 theta]
[int (3 cos theta d theta)/(27cos^3 theta) = (1/9) int(d theta)/(cos^2 theta)]
[(1/9) int (d theta)/(cos^2 theta) = (1/9) tan theta + c]
Substituting back [(sin^(-1)(x/3))] for [theta] yields:
[int (dx)/((9-x^2)^3/2)= (1/9) tan((sin^(-1)(x/3)))+ c]
Hence, evaluating the given integral yields [int (dx)/((9-x^2)^3/2)= (1/9) tan((sin^(-1)(x/3))) + c]
, using the option B. [x = 3sin theta] .
![For the indefinite integral below, choose which of the following substitutions would be most
helpful in evaluating the integral.
Enter the appropriate letter (A,B, or C) in each blank.
A. x= 3tan(theta)
B. x= 3sin(theta)
C. x= 3sec(theta)
integrate of (dx)/((9-x^2)^3/2)
Solution
Selecting the option B. [x = 3sin theta =>] [dx = 3 cos theta d theta] yields:
[int (3 cos theta d theta)/((9-9sin^2 theta)sqrt(9 - 9 sin theta))]
Factoring out 9 yields:
[int (3 cos theta d theta)/(27(1-sin^2 theta)sqrt(1 - sin^2 theta))]
Using the fundamental formula of trigonometry yields:
[1-sin^2 theta = cos^2 theta]
[int (3 cos theta d theta)/(27cos^3 theta) = (1/9) int(d theta)/(cos^2 theta)]
[(1/9) int (d theta)/(cos^2 theta) = (1/9) tan theta + c]
Substituting back [(sin^(-1)(x/3))] for [theta] yields:
[int (dx)/((9-x^2)^3/2)= (1/9) tan((sin^(-1)(x/3)))+ c]
Hence, evaluating the given integral yields [int (dx)/((9-x^2)^3/2)= (1/9) tan((sin^(-1)(x/3))) + c]
, using the option B. [x = 3sin theta] .](https://image.slidesharecdn.com/fortheindefiniteintegralbelowchoosewhichofthefollowingsub-230402104543-e162f4a1/85/For-the-indefinite-integral-below-choose-which-of-the-following-sub-pdf-1-320.jpg)
