Useful Mnemonic: DETAIL

[20120202]
A student had an interesting random thought.

What to do when the integrand is a product of three functions?

In particular he asked about $\int x e^x \cos (x)dx$.

What do you think?

[20120207]
I had you work on this in class. Please complete the problem and ask questions if you have any.

[20120202]
Well, as you should learn and commit to memory, the integration by parts formula is $$\int u dv = u\cdot v - \int v du$$ or you may know it as $$\int f(s) g^{\prime}(s) ds = f(s) g(s)- \int g(s) f^{\prime} ds$$. In any case, it deals with a product of two functions. Thus, to apply integration by parts to a product of three functions, you simply have to choose how to group your functions.

While learning how to choose $u$ and $dv$ takes practice, there's a mnemonic that might be of use: DETAIL. D stands for the differential, so the mnemonic suggests the order in which you want to choose $dv$. E stands for exponential, T stands for trigonometric, A stands for algebraic, I stands for inverse functions, and L stands for logarithmic functions.

This mnemonic would suggest we choose $u=x\cos(x)$ and $dv= e^x dx$. This contrasts with the student's choice of $u=e^x$ and $dv=x \cos(x) dx$.

I suggest you try the above two choices as well as other possible choices for $u$ and $dv$.

I will eventually link my write-up of these choices,* but I will wait and give students a chance to respond and participate.

Note: A similar mnemonic is LIATE. This would be used to determine $u$. LIATE would suggest we choose $u=x$ and $dv=e^x \cos(x) dx$.

Another Note: In general, neither DETAIL nor LIATE will guarantee useful or good choices for $u$ and $dv$. However, they provide decent starting points.

*20191222: the write-up can be found here.