Do we always have to use "LIATE" to do integration by parts?
L = logarithms
I = inverse trigonometric functions
A = algebraic functions
T = trigonometric functions
E = exponential functions
So basically, an example is a problem like$$\int -(e^x)\sin x\,dx$$
If I use the LIATE order to integrate this, I will get:
$$- \int (e^x)\sin x \,dx = e^x \sin x - e^x \cos x - \int e^x\sin x \,dx$$
If I add $\int e^x\sin x\,dx$ to both sides, I will be left with $0$ on the left-hand side.
But however, if I reverse the order —meaning, I integrate the trigonometric function $\sin x$ and differentiate the exponential part $e^x$ instead of doing the opposite— I can get an answer.
My question is:
$\endgroup$ 8My book said order was a rule. Is it really?
1 Answer
$\begingroup$It’s just a rule of thumb designed to make the integration by parts yield something that’s easy to integrate. It’s based on another rule of thumb, namely that $u$ should be easily differentiable and $dv/dx$ should be easily integrable. But choosing not to use these rules of thumb won’t get you something mathematically invalid; if violating them allows you to get the answer more efficiently then by all means do so.
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