$$\int f g \,dx=f\int g\,dx -\int f'\int g\,dx\,dx$$Note : "first" function is $f$ and "second" function is $g$.
$$\int f g h \,dx=?$$
$$\int f g h \cdots \,dx=?$$
I was trying the Fourier Cosine Transform of the function $f(x)=xe^{-ax}$ for $a>0$,
I ended up on the integral,
$$I=\int_0^\infty xe^{-ax}\cos(\lambda x)\,dx$$
Now, this has been a long time question of mine on how to extend the integration by parts rule to more than two functions,
It took me two attempts to figure out that integral by picking the "first" function as $x$ and "second" function as $e^{-ax}\cos(\lambda x)$, and applying a integration by parts again for the "second" function. My other attempt involved picking $e^{-ax}$ as "first" and $x\cos(\lambda x)$ as "second" function respectively, but this did not seem to be integrable as easily as my previously mentioned attempt.
The main motivation behind this post is since time is a constraint in exams, I am unsure of how to pick the right "first" and "second" function to apply integration by parts and get it cleared in one attempt.
Although there exists a generic rule to prioritize which function becomes "first" and "second" function, known as "ILATE" where,
- I = inverse
- L= logarithmic
- A = algebraic
- T= trigonometric
- E = exponential
But this rule is not a rigorously applied and also sometimes fails when trying by parts for more than two functions as seen my above explanation of my "other" attempt.
I am looking for advices/formula/tricks which can help prioritize (in broad sense) which function to pick as "first" and "second" and then apply by parts "recursively".
Note : I am not looking for solution of above integral