Integration Anomaly (1 Viewer)

[khfreakau]

So, in class we came across this problem, and we're still pondering the explanation. Thoughts?

The integral we want to find is that of


METHOD 1:



Which simplifies to


Now I'll leave it in this form for the sake of comparison.

METHOD 2




Now substituting in



This result is different to what was obtained using the t method formula. Of course, there's an arbitrary constant added, but how can the difference lie in the constant if there's an additional variable of t in the numerator?

 

[Drongoski]

(they differ by a mere constant)

Therefore the 2 answers are equivalent

There are many integrals with different answers when done differently but which at 1st look appear different but are in fact equivalent.

The 2 look so deceptively different, I know.

 

[khfreakau]

Indeed, it's quite interesting. I haven't encountered such an integral before where they vary in terms of a constant yet there's a different variable. The logical conclusion would be to assume that t is constant, but how would that work? We also concluded that the constant was 1, but it's quite interesting.

 

[Trebla]

t is obviously not a constant
Note that
2/(1 - t) + c
= (1 - t + 1 + t)/(1 - t) + c
= 1 + (1 + t)/(1 - t) + c
= (1 + t)/(1 - t) + k

Another example commonly encountered is something like ∫ sec²x tan x dx
∫ sec²x tan x dx
= ∫ (tan x)¹ sec²x dx
= (tan²x)/2 + c₁
OR
∫ sec²x tan x dx
= ∫ (sec x)¹(sec x tan x) dx
= (sec²x)/2 + c₂

Both are equivalent since sec²x - tan²x = 1, so
= (sec²x)/2 + c₂
= (1 + tan²x)/2 + c₂
= (tan²x)/2 + c₂ + 1/2
= (tan²x)/2 + c₁

 

[cutemouse]

If you had limits then you could also use complex analysis methods (not in the scope of the current HSC course).