In any 'manual' You can find the 'derivative' ...
(1)
... so that is...
(2)
On WolframAlpha [I wonder why...] a different and more complicated formula is reported...
Wolfram Mathematica Online Integrator
Anyway following TCM we obtain...
(3)
Kind regards
Hello, transgalactic!
Use "Trig Substitution".
Evidently, you aren't familiar with it . . .
Let:
Note that:
. . . . . . . .
Substitute: .
. . . . . .
. . . . . . .[1]
Back-substitute: .
is in a right triangle with:
Pythagorus gives us: .
Hence: .
We have: .
Substitute into [1]:
. .
. .
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
I have no idea why so many textbooks and sources use
. . which requires extra steps and looks more intimidating.
Wow!! Absolutely everyone has mis-read the a as a^2!
Not massively consequential, but an excuse for some pictures...
Then, swap the inner function...
... where (key in spoiler) ...
Spoiler:
Spoiler:
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Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
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