FP2 Integral

Find the integral of 1/(x^2+1)^2

What substitutions do you know that might help you? What would you do if the denominator were just (x^2 + 1)?

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Alternatively, you could opt for the substitution $x = \sinh u$

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I'd presume so. It's been a while since I've looked at polar coordinates. Do you have an example with which I can work?

"find the area of a single loop of the curve with equation r = acos3x"

The limits are pi/6 and -pi/6 but im not sure how they got that? thank you

This is going to be a tough one to explain.

Before looking at the polar curve, a very brief glance at the trigonometric curve should give you a better idea of what is going on. For a loop to exist in the very first place, the endpoints of the region of the angle you're looking at must give a zero value for $r$. That much should be clear. Now it's just a matter of concerning yourself only with the positive values of $r$ (as most exam boards don't bother with -ve values) for which the endpoints are equal to 0.



(I couldn't get Wolfram to graph the output as $[-a, a]$)
In this respect, you could equally find the the integral $\displaystyle \int_{\frac{\pi}{2}}^{\frac{5\pi}{6}}$ and you'd get the same area.

oh, okay! that's a lot clearer now, thank you vm!

No prob. I edited a bit of my reply to make the "region of x for which the loop is enclosed" bit a tad clearer :-)

Find the integral of 1/(x^2+1)^2

it did clarify, thank you

Nice question! Out of curiosity, did you get your final area as $\frac{a^2 \pi}{12}$ ?

yes! after finding the limits, it's really easy to do

Somebody's confident