Can you please explain this equality? int r/theta^2 exp(-r^2/(2theta^2))dr=-exp(-r^2/2theta^2)

Jairo Decker 2022-10-24 Answered
Can you please explain this equality?
r σ 2 exp ( r 2 2 σ 2 ) d r = exp ( r 2 2 σ 2 )
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Answers (1)

Ramiro Sosa
Answered 2022-10-25 Author has 13 answers
To check that an indefinite integral is equal to a function
f ( x ) d x = F ( x ) ( + C )
you can simply check that F ( x ) = f ( x ). In your example you check that
d d r [ exp ( r 2 2 σ 2 ) ] = exp ( r 2 2 σ 2 ) d d r r 2 2 σ 2 = exp ( r 2 2 σ 2 ) r σ 2
If you don't like this, then you can also find the integral using the substitution method with u = r 2 2 σ 2 . So then
d u = 2 r 2 σ 2 d r d u = r σ 2 d r
So the integral becomes
e u d u .
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