Summary

This chapter's toolkit reduces to a short list of building blocks, best remembered in groups.

Power, log, and exponential forms. $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+c$ ($n\ne-1$, the power rule) and its edge case $\int\dfrac1x\,dx=\log|x|+c$; $\int e^x\,dx=e^x+c$ and $\int a^x\,dx=\dfrac{a^x}{\log a}+c$; plus the two "do-nothing/scale" rules $\int k\,dx=kx+c$ and $\int0\,dx=c$.

Trigonometric forms. The six basic derivative pairs run in reverse to give $\int\sin x\,dx=-\cos x+c$, $\int\cos x\,dx=\sin x+c$, $\int\sec^2x\,dx=\tan x+c$, $\int\csc^2x\,dx=-\cot x+c$, $\int\sec x\tan x\,dx=\sec x+c$, and $\int\csc x\cot x\,dx=-\csc x+c$. Alongside these sit four "integrated-by-manipulation" log-form results: $\int\tan x\,dx=\log|\sec x|+c$, $\int\cot x\,dx=\log|\sin x|+c$, $\int\sec x\,dx=\log|\sec x+\tan x|+c$, and $\int\csc x\,dx=\log|\csc x-\cot x|+c$.

Inverse-trigonometric forms. $\int\dfrac{dx}{\sqrt{1-x^2}}=\sin^{-1}x+c$ and $\int\dfrac{dx}{1+x^2}=\tan^{-1}x+c$ are the two seeds from which every Type-I inverse-trig/log result of \S11.7.9 is built, simply by rescaling $x\to x/a$.

The linearity and shift rules that make everything else work. $\int kf(x)\,dx=k\int f(x)\,dx$ and $\int(f_1(x)\pm f_2(x))\,dx=\int f_1(x)\,dx\pm\int f_2(x)\,dx$ let a complicated integrand be split into pieces and handled one at a time; and if $\int f(x)\,dx=g(x)+c$, then $\int f(ax+b)\,dx=\dfrac1a g(ax+b)+c$, so any already-known antiderivative can be reused whenever the variable is replaced by a linear expression $ax+b$.

The three general integration methods. Substitution replaces $x$ by a new variable to turn an unfamiliar integrand into a recognisable one (or, in the $\int f'(x)[f(x)]^n\,dx$ / $\int\frac{f'(x)}{f(x)}\,dx$ shapes, is spotted directly without writing out a formal substitution). Decomposition rewrites an integrand -- most often a rational function -- as a sum of simpler pieces, via partial fractions or an algebraic/trigonometric identity, before integrating term by term. Integration by parts, $\int u\,dv=uv-\int v\,du$, handles a product of two dissimilar functions (polynomial-times-trig, polynomial-times-exponential, or a lone log/inverse-trig factor); repeated application is systematised by Bernoulli's rule $\int u\,dv=uv-u'v_1+u''v_2-\cdots$, whose two signature results are the self-referencing pair

$\int e^{ax}\sin bx\,dx=\frac{e^{ax}}{a^2+b^2}[a\sin bx-b\cos bx]+c,\qquad \int e^{ax}\cos bx\,dx=\frac{e^{ax}}{a^2+b^2}[a\cos bx+b\sin bx]+c.$ …