How does one carry out
$\displaystyle \int\frac{1}{(x^2+1)^2}dx$
- Wolfram|Alpha
Click on 'show steps'
Edit:
Just in case a picture helps...
... where
... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which subject to the chain rule).
Carry on the anti-clockwise journey (to I) by finding G(theta) and mapping back from tan(theta) to x.
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Don't integrate - balloontegrate!
http://www.ballooncalculus.org/examples/gallery.html
http://www.ballooncalculus.org/asy/doc.html