# 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=...

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)

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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` .**