Is it worth while to learn partial fraction decomposition and reduction formula or do most mathematicians rely on computers for solving integrals?
I’ve just finished Calculus I but not sure if I will go on to take Calculus II. If partial fraction decomp/reduction formula is worth becoming proficient at solving by hand, does anyone have any recommended resources for learning these mentioned techniques?
Computers are definitely better in actually calculating partial fractions than humans, but mathematics is not calculating but understanding! I tell my students that mathematics is the art to avoid calculations.
Concerning partial fractions, I had a funny/frustrating experience some weeks ago. In the anaysis lecture I presented the general existence theorem together with my favorit method to calculate the coefficients: If one has an $n$-fold zero $x_0$ in the denominator of $r(x)=p(x)/q(x)$ then the partial fractions representation contains the term $c_k/(x-x_0)^k$ for $k=1,\ldots,n$ and you can determine the coefficients by calculating $\tilde r(x)=(x-x_0)^n r(x)$ (you erase the factor $(x-x_0)^n$ in the denominator) to get $c_n=\tilde r(x_0)$, $c_{n-1}=\tilde r'(x_0)$, $c_{n-2}=\tilde r''(x_0)/2$ and so on.
In the exam there was then a concrete example. Less than half of students used the propesed method and almost all of them got the correct result easily and quickly. Unfortunately, the majority of the students did not trust me and prepared themselves with some sources from the internet and youtube tutorials which almost all suggest to find the $c_k$ by comparing the coefficients of $p(x)$ and $q(x)$ times the partial fractions deomposition. None of them got the correct result!
