My method is similar to yours. DI table may make the integration by parts neater.I have a feeling this is not the fastest method...
(the proof is left as an exercise to the reader.)
From here, I will use the result , which is not difficult to verify via the appropriate double angle identity.
Note that the absolute value is important when evaluating the lower bound.
As it turns out, the indefinite counterparts of some of these integrals have no closed-form solutions (I tried googling a solution to no.9), so some of these definite integrals must be solved (assuming they can be solved conventionally) without directly integrating into the anti-derivative, similar to how the integral of exp(x)cos(x) is done by using integration by parts twice (so that we have 2*integral = something).
These are Mr Blyatman's question. Ill work them up later today
After dividing top and bottom by 27^x, the integral is in the form 1/(p^x+q^x+r^x). If this integral has an elementary form, then there must be a relationship between p,q and r. After some trial and error, you would get that relationship.How are you supposed to know phi is a good sub here? I mean looking at the original question alone
That... doesn't necessarily have to be true, but my intuition agrees with that.If this integral has an elementary form, then there must be a relationship between p,q and r. After some trial and error, you would get that relationship.