funkinvile
New member
- Joined
- Nov 2, 2015
- Messages
- 2
< link to objectionable page removed >
I used this substitution:
\(\displaystyle x\, =\, \tan(\theta)\)
\(\displaystyle dx\, =\, \sec^2(\theta)\, d\theta\)
Then I did:
\(\displaystyle \displaystyle \int\, \dfrac{1}{x^2\, \sqrt{\strut\, 1\, +\, x^2\,}}\, dx\, =\, \int\, \dfrac{\sec^2(\theta)}{x^2\, \sqrt{\strut\, 1\, +\, \tan^2(\theta)\,}}\, d\theta\, =\, \int\, \dfrac{\sec^2(\theta)}{x^2\, \sqrt{\strut\, \sec^2(\theta)\,}}\, d\theta\, =\, \int\, \dfrac{\sec^2(\theta)}{\tan^2(\theta)\, \sec(\theta)}\, d\theta\)
\(\displaystyle \displaystyle =\, \int\, \dfrac{\sec(\theta)}{\tan^2(\theta)}\, d\theta\, =\, \int\, \dfrac{\left(\dfrac{1}{\cos(\theta)}\right)}{\left(\dfrac{\sin(\theta)}{\cos(\theta)}\right)}\, d\theta\, =\, \int\, \left(\dfrac{1}{\cos(\theta)}\right)\, \left(\dfrac{\cos(\theta)}{\sin(\theta)}\right)\, d\theta\, =\, \int\, \csc(\theta)\, d\theta\)
On this <reference removed>, I have a picture of my paper work and a picture of the solution manual to the problem I am trying to solve.
My question is: how is it that the solutions manual uses "1/cos(x)" and ends up with the integral of "cos(x)/sin^2(x)" whereas when I use "sec(x)" and do the math I end up with the integral of "csc(x)"?
Please tell me where I went wrong I am so dang frustrated
I used this substitution:
\(\displaystyle x\, =\, \tan(\theta)\)
\(\displaystyle dx\, =\, \sec^2(\theta)\, d\theta\)
Then I did:
\(\displaystyle \displaystyle \int\, \dfrac{1}{x^2\, \sqrt{\strut\, 1\, +\, x^2\,}}\, dx\, =\, \int\, \dfrac{\sec^2(\theta)}{x^2\, \sqrt{\strut\, 1\, +\, \tan^2(\theta)\,}}\, d\theta\, =\, \int\, \dfrac{\sec^2(\theta)}{x^2\, \sqrt{\strut\, \sec^2(\theta)\,}}\, d\theta\, =\, \int\, \dfrac{\sec^2(\theta)}{\tan^2(\theta)\, \sec(\theta)}\, d\theta\)
\(\displaystyle \displaystyle =\, \int\, \dfrac{\sec(\theta)}{\tan^2(\theta)}\, d\theta\, =\, \int\, \dfrac{\left(\dfrac{1}{\cos(\theta)}\right)}{\left(\dfrac{\sin(\theta)}{\cos(\theta)}\right)}\, d\theta\, =\, \int\, \left(\dfrac{1}{\cos(\theta)}\right)\, \left(\dfrac{\cos(\theta)}{\sin(\theta)}\right)\, d\theta\, =\, \int\, \csc(\theta)\, d\theta\)
On this <reference removed>, I have a picture of my paper work and a picture of the solution manual to the problem I am trying to solve.
My question is: how is it that the solutions manual uses "1/cos(x)" and ends up with the integral of "cos(x)/sin^2(x)" whereas when I use "sec(x)" and do the math I end up with the integral of "csc(x)"?
Please tell me where I went wrong I am so dang frustrated
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