（一）含有 $ax+b$ 的积分 $\ \\\displaystyle1.\int\frac{{\rm d}x}{ax+b}=\frac1a\ln|ax+b|+C.\\ 2.\int(ax+b)^\mu{\rm d}x=\frac1{a(\mu+1)}(ax+b)^{\mu+1}+C\;(\mu\ne-1).\\ 3.\int\frac x{ax+b}{\rm d}x=\frac1{a^2}(ax+b-b\ln|ax+b|)+C.\\ 4.\int\frac{x^2}{ax+b}{\rm d}x=\frac1{a^3}\left[\frac12(ax+b)^2-2b(ax+b)+b^2\ln|ax+b|\right]+C.\\ 5.\int\frac{{\rm d}x}{x(ax+b)}=-\frac1b\ln\left|\frac{ax+b}x\right|+C.\\ 6.\int\frac{{\rm d}x}{x^2(ax+b)}=-\frac1{bx}+\frac a{b^2}\ln\left|\frac{ax+b}x\right|+C.\\ 7.\int\frac x{(ax+b)^2}{\rm d}x=\frac1{a^2}\left(\ln|ax+b|+\frac b{ax+b}\right)+C.\\ 8.\int\frac{x^2}{(ax+b)^2}{\rm d}x=\frac1{a^3}\left(ax+b-2b\ln|ax+b|-\frac{b^2}{ax+b}\right)+C.\\ 9.\int\frac{{\rm d}x}{x(ax+b)^2}=\frac1{b(ax+b)}-\frac1{b^2}\ln\left|\frac{ax+b}x\right|+C.\\$ （二）含有 $\sqrt{ax+b}$ 的积分 $\ \\\displaystyle10.\int\sqrt{ax+b}\,{\rm d}x=\frac2{3a}\sqrt{(ax+b)^3}+C.\\ 11.\int x\sqrt{ax+b}\,{\rm d}x=\frac2{15a^2}(3ax-2b)\sqrt{(ax+b)^3}+C.\\ 12.\int x^2\sqrt{ax+b}\,{\rm d}x=\frac2{105a^3}\left(15a^2x^2-12abx+8b^2\right)\sqrt{(ax+b)^3}+C.\\ 13.\int\frac x{\sqrt{ax+b}}{\rm d}x=\frac2{3a^2}(ax-2b)\sqrt{ax+b}+C.\\ 14.\int\frac{x^2}{\sqrt{ax+b}}{\rm d}x=\frac2{15a^3}\left(3a^2x^2-4abx+8b^2\right)\sqrt{ax+b}+C.\\ 15.\int\frac{{\rm d}x}{x\sqrt{ax+b}}= \begin{cases} \displaystyle\frac1{\sqrt b}\ln\left|\frac{\sqrt{ax+b}-\sqrt b}{\sqrt{ax+b}+\sqrt b}\right|+C\;(b>0),\\\\ \displaystyle\frac2{\sqrt{-b}}\arctan\sqrt\frac{ax+b}{-b}+C\;(b<0). \end{cases}\\ 16.\int\frac{{\rm d}x}{x^2\sqrt{ax+b}}=-\frac{\sqrt{ax+b}}{bx}-\frac a{2b}\int\frac{{\rm d}x}{x\sqrt{ax+b}}.\\ 17.\int\frac{\sqrt{ax+b}}x{\rm d}x=2\sqrt{ax+b}+b\int\frac{{\rm d}x}{x\sqrt{ax+b}}.\\ 18.\int\frac{\sqrt{ax+b}}{x^2}{\rm d}x=-\frac{\sqrt{ax+b}}x+\frac2a\int\frac{{\rm d}x}{x\sqrt{ax+b}}.\\$ （三）含有 $x^2\pm a^2$ 的积分 $\ \\\displaystyle19.\int\frac{{\rm d}x}{x^2+a^2}=\frac1a\arctan\frac xa+C.\\ 20.\int\frac{{\rm d}x}{(x^2+a^2)^n}=\frac x{2(n-1)a^2(x^2+a^2)^{n-1}}+\frac{2n-3}{2(n-1)a^2}\int\frac{{\rm d}x}{(x^2+a^2)^{n-1}}.\\ 21.\int\frac{{\rm d}x}{x^2-a^2}=\frac1{2a}\ln\left|\frac{x-a}{x+a}\right|+C.\\$ （四）含有 $ax^2+b\;(a>0)$ 的积分 $\ \\\displaystyle22.\int\frac{{\rm d}x}{ax^2+b}= \begin{cases} \displaystyle\frac1{\sqrt{ab}}\arctan\sqrt\frac abx+C\;(b>0),\\\\ \displaystyle\frac1{2\sqrt{-ab}}\ln\left|\frac{\sqrt ax-\sqrt{-b}}{\sqrt ax+\sqrt{-b}}\right|+C\;(b<0). \end{cases}\\ 23.\int\frac x{ax^2+b}{\rm d}x=\frac1{2a}\ln\left|ax^2+b\right|+C.\\ 24.\int\frac{x^2}{ax^2+b}{\rm d}x=\frac xa-\frac ba\int\frac{{\rm d}x}{ax^2+b}.\\ 25.\int\frac{{\rm d}x}{x(ax^2+b)}=\frac1{2b}\ln\frac{x^2}{|ax^2+b|}+C.\\ 26.\int\frac{{\rm d}x}{x^2(ax^2+b)}=-\frac1{bx}-\frac ab\int\frac{{\rm d}x}{ax^2+b}.\\ 27.\int\frac{{\rm d}x}{x^3(ax^2+b)}=\frac a{2b^2}\ln\frac{\left|ax^2+b\right|}{x^2}-\frac1{2bx^2}+C.\\ 28.\int\frac{{\rm d}x}{(ax^2+b)^2}=\frac x{2b(ax^2+b)}+\frac1{2b}\int\frac{{\rm d}x}{ax^2+b}.\\$ （五）含有 $ax^2+bx+c\;(a>0)$ 的积分 $\ \\\displaystyle29.\int\frac{{\rm d}x}{ax^2+bx+c}= \begin{cases} \displaystyle\frac2{\sqrt{4ac-b^2}}\arctan\frac{2ax+b}{\sqrt{4ac-b^2}}+C\;(b^2<4ac),\\\\ \displaystyle\frac1{\sqrt{b^2-4ac}}\ln\left|\frac{2ax+b-\sqrt{b^2-4ac}}{2ax+b-\sqrt{b^2-4ac}}\right|+C\;(b^2>4ac). \end{cases}\\ 30.\int\frac x{ax^2+bx+c}{\rm d}x=\frac1{2a}\ln\left|ax^2+bx+c\right|-\frac b{2a}\int\frac{{\rm d}x}{ax^2+bx+c}.\\$ （六）含有 $\sqrt{x^2+a^2}\;(a>0)$ 的积分 $\ \\\displaystyle31.\int\frac{{\rm d}x}{\sqrt{x^2+a^2}}=\mathrm{arsh}\,\frac xa+C_1=\ln(x+\sqrt{x^2+a^2})+C.\\ 32.\int\frac{{\rm d}x}{\sqrt{(x^2+a^2)^3}}=\frac x{a^2\sqrt{x^2+a^2}}+C.\\ 33.\int\frac x{\sqrt{x^2+a^2}}{\rm d}x=\sqrt{x^2+a^2}+C.\\ 34.\int\frac x{\sqrt{(x^2+a^2)^3}}{\rm d}x=-\frac1{\sqrt{x^2+a^2}}+C.\\ 35.\int\frac{x^2}{\sqrt{x^2+a^2}}{\rm d}x=\frac x2\sqrt{x^2+a^2}-\frac{a^2}2\ln(x+\sqrt{x^2+a^2})+C.\\ 36.\int\frac{x^2}{\sqrt{(x^2+a^2)^3}}{\rm d}x=-\frac x{\sqrt{x^2+a^2}}+\ln(x+\sqrt{x^2+a^2})+C.\\ 37.\int\frac{{\rm d}x}{x\sqrt{x^2+a^2}}=\frac1a\ln\frac{\sqrt{x^2+a^2}-a}{|x|}+C.$

未完待续……