The original poster encounters a challenging integral involving a square root and a rational function, specifically \(\int^\sqrt{3}_1 \frac{dx}{\sqrt{(1+x^2)^3}}\). The discussion revolves around finding an appropriate substitution to simplify the integral.
Participants are actively engaging with the problem, with one providing a suggestion for a trigonometric substitution. The original poster is working through the implications of this substitution and has identified a mistake in their calculations. There is a collaborative effort to clarify the steps involved.
There are indications of confusion regarding the transformation of the integral after substitution, and participants are questioning the correctness of the steps taken. The original poster is also navigating through the limits of integration as part of the substitution process.
tonit said:I'm now stuck at this point:
[itex]\int^\frac{\pi}{3}_\frac{\pi}{4} \frac{dt}{cos^3(t)}[/itex]
tonit said:x = tan(t)
dx = d(tan[t]) = [itex]\frac{dt}{cos^2(t)}[/itex]
x = 1 => tan(t) = 1 => t = [itex]\frac{\pi}{4}[/itex]
x = [itex]\sqrt{3}[/itex] => tan(t) = [itex]\sqrt{3}[/itex] => t = [itex]\frac{\pi}{3}[/itex]
so the integral becomes
[itex]\int^\frac{\pi}{3}_\frac{\pi}{4} \frac{dt}{cos^2(t)\sqrt{(1+tan^2(t))^3}}[/itex]
oh and now I spot my mistake. It will become cos(t)dt