How to evaluate base indefinite integrals

Question

I can find a list of known integrals anywhere, but how would I build this list myself? For example how can I prove that $\int x^2 dx = x^3 / 3$? I want to understand the general theory. It would be good to have some links to detailed explanations. Thanks.

Answer 1

First you should understand what $\int f = g$ means. On the left you have an indefinite integral which has nothing to do with integrals, instead it means that $g$ is a primitive (antiderivative) of $f$ i.e. $g'=f$. So, if you know how to compute derivatives, you can easily check such equalities.

The connection between definite integrals and antiderivatives is given by the fundamental theorem of calculus. That theorem justifies the notation used to represent antiderivatives.

Answer 2

A list of known integrals ist most easily obtained from fplaying around with diferentiation. After all, taking derivatives is way more "mechanical" than integrating. Thus if you know that the derivative of $x^n$ ($n\ne0$) is $nx^{n-1}$, you get the integration rule $ \int nx^{n-1}\,\mathrm dx=x^n+C$, which after dividing by $n\ne0$ and replacing$n$ with $n+1$ becomes $$ \int x^n\,\mathrm dx=\frac1{n+1}x^{n+1}+C\qquad\text{if }n\ne-1.$$ Of course, this leaves us with the riddle of what is $\int \frac{\mathrm dx}x$, and that will only be solved "by chance" when you keep filling up your table using differentiation results for standard functions and arrive at taking the derivative of $\ln x$ to find $\frac1x$ and thus add $$ \int x^{-1}\,\mathrm dx=\ln x+C$$ to your list. Proceeding like this quickly gives you a fairly large list of basic integrals. Admittedly, some "real" art of integration may be nedcessary to extend the list (e.g. use integration by parts to find $\int\ln x\,\mathrm dx$)