Other way to solve this is to put just one substitution so that the problem can be solved quickly.
If we put the integral becomes and we're done.
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The key is to turn that into an immediate integral; when we set we have a 50% of the work done, and then we need to get our arctagent, so we again put and then combine these substitutions to produce to get a faster result. (Apply the same procedure to your other integral.)
One thing it's good for is handling substitutions that enable us to work backwards through the chain rule. On the other hand, we often do quite well without it differentiating - and say, e.g.,
And mapping out the pattern...
... we can often manage equally well without dy/dx when we're travelling 'up' (i.e. integrating)...
Don't integrate - balloontegrate!
Balloon Calculus Forum
None taken. I shouldn't have left out my usual spiel:
"straight continuous lines differentiate downwards (integrate up) with respect to the main variable, the straight dashed line similarly but with respect to the dashed balloon expression."
But I'd be grateful to know if you'd perused some other examples without enlightenment.
Hey - you're expert in the notations you use, so perhaps it's not for you...
Cheers - but take another look!
Tom