Howdy,
So, i have a question about partial integration.
Acctually this question concerns Integration in general.
How do you know how to choose the variables for substitution?
For example i have
If we pick U = and V =
dU = and dV =
so...
=
...
This is the point where i start to doubt wether i've choosen the right variables for substitution.
"Integration by parts", not "partial integration".
You try both ways. Obviously if you are going to go from u to du and from dv to v, you will want to choose u to be easily differentiable and dv to easily integrable. Also, you will then need to integrate vdu so you want that to be simpler than "udv". Here, and are easily both differentiable and integrable. So you look at "udv".Acctually this question concerns Integration in general.
How do you know how to choose the variables for substitution?
For example i have
If you try and you have du= 2xdx and so . Yes, that's easier because the power of x has been reduced. I think I would try integration by parts again!
If you try u= cos(x) and you have and so that . Power of x has gone up one- more complicated!
If we pick U = and V =
dU = and dV =
so...
=
...
This is the point where i start to doubt wether i've choosen the right variables for substitution.
L.I.A.T.E. (logs, inverse trigs, algebra, trigs, exponentials) Integration by parts - Wikipedia, the free encyclopedia seems a handy thing.
However, if a certain amount of trial and error is inevitable, then, with no disrespect to the honorable tradition of 'vdu and udv', I find it can help to work directly with the expressions thus denoted. Then the choice is a matter of depicting the product-rule with 'legs crossed or uncrossed'.
The challenge of integration by parts is simply to fill out one of these shapes usefully, but starting in one of the bottom corners instead of (as happens in differentiation) the top.
As has been pointed out, you were right on track, because...
leads to...
... the only problem, that the lower level is no longer equal to what we wanted to integrate...
But fix that...
Now we face a problem similar to the first one, so it's a case of 'integration by parts twice'...
... or cutting out the l.h.s. for a bit more space...
Speaking for myself, if I take the wrong road of...
... leading to ...
... then I'm much better able to review and re-attack than if I'm juggling u's and v's. (OR... I'll use a diagram to help me re-assign them.)
Don't integrate - balloontegrate!
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