Are there examples of when the ILATE mnemonic for choosing factors when integrating by parts fails?

Question

Is it possible in some cases that using the ILATE rule does not yield an explicit antiderivative but making another choice does yields one? If so, please give examples.

Answer 1

Yes, it's easy for the rule to fail if the proposed derivative is not integrable. For example in the integral

$$\int x^3 e^{x^2} \mathrm{d}x$$

the rule would propose $u=x^3$ and $dv=e^{x^2}$. The latter cannot be integrated and you are therefore stuck.

To solve the above integral use $u=x^2$ and $dv=x e^{x^2}$ instead. Then you get $du=2x$ and $v=\frac{1}{2}e^{x^2}$, leading to

$$\int x^3 e^{x^2} \mathrm{d}x=\\ \frac{x^2}{2}e^{x^2} -\int 2x \frac{1}{2}e^{x^2} \mathrm{d}x = \\ \frac{x^2}{2}e^{x^2} - \frac{1}{2}e^{x^2}=\\ \frac{1}{2}e^{x^2}(x^2-1)$$

There are likely other types of examples, but I can't think of any at the moment.

Answer 2

Another example is $\int\frac{xe^x}{(x+1)^2} dx$.

According to this rule, you would let $u=\frac{x}{(x+1)^2}$ and $dv=e^x dx$,

which would give $\frac{xe^x}{(x+1)^2}-\int\frac{(1-x)e^x}{(x+1)^3}dx$.

Instead, you want to let $u=xe^x$ and $dv=\frac{1}{(x+1)^2}dx$,

which gives $-\frac{xe^x}{x+1}-\int -e^x dx$.