Do we need to memorise this identity? (2 Viewers)

[leehuan]

Yeah if the constant marks the greatest number of a Pythagorean triple it may be faster to use the auxiliary angle method.

 

[Paradoxica]

That's interesting. I have not investigated, but I noticed that in both integrals, {3,4,5} and {5,12,13} are Pythagorean Triples.

It's because of the Auxiliary Transformation takes √_a²+b² for the harmonic coefficient, but since it is a Pythagorean Triple, that coefficient is, by definition, c.

 

[Paradoxica]

Yeah if the constant marks the greatest number of a Pythagorean triple it may be faster to use the auxiliary angle method.

But in any case, it doesn't need to be a Pythagorean Triple.

As long as asinx + bcosx + c satisfies a²+b²=c², the Auxiliary Transformation and Tangent Half Angle Substitution are guaranteed to be nice.

 

[leehuan]

But in any case, it doesn't need to be a Pythagorean Triple.

As long as asinx + bcosx + c satisfies a²+b²=c², the Auxiliary Transformation and Tangent Half Angle Substitution are guaranteed to be nice.

Yeah basically point is to use one or the other