At my undergraduate institution (Facultad de Ciencias, UNAM, Mexico), for a while in the mid-to-late 70s, several professors in the Calculus sequence (four courses: Differential single-variable (Calc I), Integral single-variable (Calc II), Differential multi-variable (Calc III), Integral multi-variable (Calc IV)) decided to use Hasse's analysis textbook instead of a calculus textbook. It was more of a "baby analysis" than a calculus course.
Now, this was done only in courses that were being taught to Math, Actuarial Sciences, and Physics majors (and a Math major takes nothing but math courses, for instance).
It did not go well. Students didn't learn analysis very well, and they certainly did not learn the calculus skills they needed very well. The Physics department, in particular, went up in arms because the Physics majors were coming out of these courses unable to actually compute integrals and derivatives, or use them to solve specific physics problems. Same problem with the actuarial scientists. The math majors fared a little better, but mainly because the same people who were doing this were the people who were also teaching the analysis courses in the junior and senior years; but those that went on to take analysis from other people didn't do so well. In addition, the failure rate for these courses was extremely high. (Failure rate in the Calculus sequence has always been way too high there, but it got much worse).
Most professors switched back to calculus books and to not do baby analysis. By the mid-80s, almost nobody was using Hasse's book or teaching "mini-analysis."
If a student has had a good enough calculus course in High School, then it is likely that a baby analysis course might indeed be beneficial, building on the bases that calculus can help set. This could very well be the case in the EU; it's not the case in the US. (In Mexico, nominally, students in the Math/Physics/Engineering track were taking a year of Calculus as seniors in High School, but obviously not good enough).
I don't feel I could do justice to all the possibilities in a single post. However, I do know a very informative website that covers several such errors, and moreover covers non-technical problems that students encounter.
Here is the site: I hope you find it useful!
Best Answer
Memorize the derivatives of $x^n$, $e^x$, $\ln|x|$, $\sin x$, $\cos x$, $\arcsin x$, $\arctan x$, and maybe $\tan x$ (which are used all the time), and derive the rest whenever you need them (which isn't often, in my experience).
Then many of the integrals will just be “backwards versions” of what you already know, so there's no extra memory required to store them, and for those that are not, you can compute them when needed rather than memorizing them. For example, $$ \int \ln x \, dx = \int 1 \cdot \ln x \, dx = \cdots \quad \text{(integrate by parts)} $$ or $$ \int \tan x \, dx = \int \frac{\sin x}{\cos x} \, dx = - \int \frac{-\sin x}{\cos x} \, dx = - \ln|\cos x| + C \qquad \text{(pattern recognition, $\tfrac{f'(x)}{f(x)}$)} . $$ One tricky case, which I would recommend memorizing (even though it's not included in your list) is $$ \int \frac{dx}{\sqrt{a^2+x^2}} = \ln\left|x + \sqrt{a^2+x^2}\right| + C . $$