Hello,
I have a really basic question. I read the criteria that "see Weinberg, (3.5.13-14)" refers to, and to me, it seems that the equation will always be greater than 0 since it is squared. If so, what is it's purpose?
Source...
Remember the integration constant C.
Here is what I get:##\frac{1}{3} \int sec^3(\theta) d\theta = \frac{1}{6} (9y \cdot \sqrt{(\frac{1}{3})^2 + y^2}## ##+ \ln \left|3 \big(\sqrt{(\frac{1}{3})^2 + y^2} + y) \right|)##+ C
Thanks.
In that case, I would solve it the following way:
##\int \sqrt{(1+(3y)^2)} dy \ and \ 3y = tan(\theta) \ and \ dy = \frac{sec^2(\theta) d\theta}{3}##
This gives ##\int \sqrt{(1 + tan^2(\theta))} \cdot \frac{sec^2(\theta)}{3} d\theta## or ## \frac {1}{3} \cdot \int sec^3(\theta)...
I am a bit confused. Why are people substituting ##3y = tan(\theta)##
I would substitute ##y=tan(\theta)##, and the deivative is ##\frac{dy}{dx} = tan^2(\theta) + 1 = sec^2(\theta)##. The final setup ## dy = sec^2(\theta) d \theta##.
##\int \sqrt(1 + y^2) dy = \int \sqrt(1 + tan^2(\theta))...