Calculus

1. Limit

   $\begin{array}{l}\lim (x+y-z)=\lim x+\lim y-\lim z\\ \lim \left(xyz\right)=\lim x\lim y\lim z\\ \lim \left(\frac{x}{y}\right)=\frac{\lim x}{\lim y}\\ \underset{x\to 0}{lim}{\left[Cf\right(x\left)\right]}^n=C\underset{x\to 0}{lim}{\left[f\right(x\left)\right]}^n\\ \underset{x\to \alpha }{lim}{\left[Cf\right(x\left)\right]}^n=C\underset{x\to \alpha }{lim}{\left[f\right(x\left)\right]}^n\\ \underset{x\to \infty }{lim}{e}^x=\infty \\ \underset{x\to -\infty }{lim}{e}^x=0\\ \underset{x\to 0}{lim}{a}^x=1\\ \underset{x\to \infty }{lim}\ln x=\infty \\ \underset{x\to \infty }{lim}\frac{c}{x^n}=0(n>0)\\ \underset{x\to \infty }{lim}\frac{x}{\sqrt[x]{x!}}=e\\ \underset{x\to \infty }{lim}{(1+\frac{k}{x})}^x=e^k(e=2.71)\\ \underset{x\to \infty }{lim}{(1-\frac{1}{x})}^x=\frac{1}{e}\\ \underset{x\to \infty }{lim}\frac{x^n}{x!}=0\\ \underset{x\to \infty }{lim}x{(\frac{\sqrt{2\pi x}}{x!})}^{\frac{1}{x}}=e\\ \underset{x\to \infty }{lim}\frac{x!}{x^x{e}^{-x}\sqrt{x}}=\sqrt{2\pi }\\ \underset{x\to \infty }{lim}{log}_a{(1+\frac{1}{x})}^x={log}_{a}e\\ \underset{x\to 0}{lim}\frac{{log}_e(1+x)}{x}=1\\ \underset{x\to 0}{lim}\frac{x}{{log}_a(1+x)}={log}_{a}e\\ \underset{x\to 0}{lim}\frac{a^x-1}{x}=lna(a>0)\\ \underset{x\to 0}{lim}\frac{sinx}{x}=1\\ \underset{x\to 0}{lim}\frac{tanx}{x}=1\\ \underset{x\to 0}{lim}\frac{1-cosx}{x}=0\\ \underset{x\to 0}{lim}\frac{1-cosx}{x^2}=\frac{1}{2}\\ \underset{x\to 0}{lim}\frac{arcsinx}{x}=1\\ \underset{x\to 0}{lim}\frac{arctanx}{x}=1\\ \underset{x\to 1}{lim}\frac{{(arccosx)}^2}{1-x}=2\end{array}$

2. Derivative

   $\begin{array}{l}{(Cu)}^{\prime }=C{u}^{\prime }\\ {(u+v-w)}^{\prime }=u^{\prime }+v^{\prime }-w^{\prime }\\ {(uv)}^{\prime }=u{v}^{\prime }+v{u}^{\prime }\\ {(uvw)}^{\prime }=u^{\prime }vw+v^{\prime }uw+w^{\prime }uv\\ {(\frac{u}{v})}^{\prime }=\frac{u^{\prime }v-v^{\prime }u}{v^2}\\ {[f(u(x))]}^{\prime }=f^{\prime }(u)u^{\prime }(x)\\ {(C)}^{\prime }=0\\ {(x)}^{\prime }=1\\ {(x^n)}^{\prime }=n{x}^{n-1}\\ {(\frac{1}{x})}^{\prime }=-\frac{1}{x^2}\\ {\sqrt{x}}^{\prime }=\frac{1}{2\sqrt{x}}\\ {(\ln x)}^{\prime }=\frac{1}{x}\\ {({\log }_{a}x)}^{\prime }=\frac{1}{x}{\log }_{a}e\\ {(e^x)}^{\prime }=e^x\\ {(a^x)}^{\prime }=a^x\ln a\\ {(\sin x)}^{\prime }=\cos x\\ {(\cos x)}^{\prime }=-\sin x\\ {(\tan x)}^{\prime }=\frac{1}{{\cos }^{2}x}\\ {(\mathrm{cottan}x)}^{\prime }=-\frac{1}{{\sin }^{2}x}\\ {(\arcsin x)}^{\prime }=\frac{1}{\sqrt{1-x^2}}\\ {(\arccos x)}^{\prime }=-\frac{1}{\sqrt{1-x^2}}\\ {(\arctan x)}^{\prime }=\frac{1}{1+x^2}\\ {(\mathrm{arccot}x)}^{\prime }=-\frac{1}{1+x^2}\\ {(u^v)}^{\prime }=v{u}^{v-1}{(u)}^{\prime }+u^v\ln u{(v)}^{\prime }\end{array}$

3. Differential

   $\begin{array}{l}\mathrm{dy}=y^{\prime }\mathrm{dx}\\ d(Cu)=C\mathrm{du}\\ d(u+v-w)=\mathrm{du}+\mathrm{dv}-\mathrm{dw}\\ d(uv)=u\mathrm{dv}+v\mathrm{du}\\ d(uvw)=(vw)du+(uw)dv+(uv)dw\\ d(\frac{u}{v})=\frac{vdu-udv}{v^2}\\ d(u^n)=n{u}^{n-1}du\\ d(\sin u)=\cos udu\\ d(\cos u)=-\sin udu\end{array}$

4. Indefinite integral

   $\begin{array}{l}\int f(x)\mathrm{dx}=F(x)+C\\ F^{\prime }(x)=f(x)\\ \int f(x)\mathrm{dx}=F(x)+C\\ \int \mathrm{dx}=x+C\\ \int kf(x)\mathrm{dx}=k\int f(x)\mathrm{dx}\\ \int f(u+v+w)\mathrm{dx}=\int u\mathrm{dx}+\int v\mathrm{dx}+\int w\mathrm{dx}\\ \int u\mathrm{dv}=\mathrm{uv}-\int v\mathrm{du}\\ \int f(\mathrm{kx})\mathrm{dx}=\frac{1}{k}\int f(x)\mathrm{dx}\\ \int x^m\mathrm{dx}=\frac{x^{m+1}}{m+1}+C(m\ne 1)\\ \int {(ax+b)}^n\mathrm{dx}=\frac{{(ax+b)}^{n+1}}{a(n+1)}+C(n\ne 1)\\ \int e^x\mathrm{dx}=e^x+C\\ \int k^x\mathrm{dx}=\int e^{x\ln k}\mathrm{dx}=\frac{e^{x\ln k}}{\ln k}=\frac{k^x}{\ln k}(k>0,k\ne 1)\end{array}$

5. Root integral

   $\begin{array}{l}\int \frac{\mathrm{dx}}{\sqrt{ax+b}}=\frac{2}{a}\sqrt{ax+b}+C\\ \int \sqrt{ax+b}\mathrm{dx}=\frac{2}{3a}{(ax+b)}^{\frac{3}{2}}+C\\ \int \frac{x\mathrm{dx}}{\sqrt{ax+b}}=\frac{2(ax-2b)}{3a^2}\sqrt{ax+b}+C\\ \int \frac{x\mathrm{dx}}{\sqrt{ax+b}}=\frac{2(3ax-2b)}{15a^2}{(ax+b)}^{\frac{2}{3}}+C\\ \int \frac{\mathrm{dx}}{(x+c)\sqrt{ax+b}}=\frac{1}{\sqrt{b-ac}}\ln |\frac{\sqrt{ax+b}-\sqrt{b-ac}}{\sqrt{ax+b}+\sqrt{b-ac}}|+C(b-ac>0)\\ \int \frac{\mathrm{dx}}{(x+c)\sqrt{ax+b}}=\frac{1}{\sqrt{ac-b}}\arctan \sqrt{\frac{ax+b}{ac-b}}+C(ac-b>0)\\ \int x^2\sqrt{a+bx}\mathrm{dx}=\frac{2(8a^2-12abx+15b^2)\sqrt{{(a+bx)}^3}}{105b^3}+C\\ \int \frac{\mathrm{dx}}{\sqrt{a^2+x^2}}=\ln (x+\sqrt{x^2+a^2})+C\\ \int \frac{\mathrm{dx}}{\sqrt{x^2-a^2}}=\ln (x+\sqrt{x^2-a^2})+C\\ \int \frac{\mathrm{dx}}{\sqrt{a^2-x^2}}={\sin }^{-1}\frac{x}{a}+C\end{array}$

6. Trigonometric integral

   $\begin{array}{l}\int \sin x\mathrm{dx}=-\cos x+C\\ \int \cos x\mathrm{dx}=\sin x+C\\ \int \tan x\mathrm{dx}=\ln \sec x+C=-\ln \cos x+C\\ \int \cot x\mathrm{dx}=\ln \sin x+C\\ \int {\sin }^{2}x\mathrm{dx}=\frac{x}{2}-\frac{1}{4}\sin 2x+C\\ \int {\cos }^{2}x\mathrm{dx}=\frac{x}{2}+\frac{1}{4}\sin 2x+C\\ \int {\tan }^{2}x\mathrm{dx}=\tan x-x+C\\ \int {\cot }^{2}x\mathrm{dx}=-\cot x-x+C\\ \int {\sin }^{3}x\mathrm{dx}=\frac{1}{3}{\cos }^3-\cos x+C\\ \int {\cos }^{3}x\mathrm{dx}=\sin -\frac{1}{3}{\sin }^3+C\\ \int \frac{\mathrm{dx}}{{\sin }^{2}x}=-\mathrm{cottan}x+C\\ \int \frac{\mathrm{dx}}{{\cos }^{2}x}=\tan x+C\end{array}$

7. Partial fractional integral

   $\begin{array}{l}\int \frac{\mathrm{dx}}{x}=\ln |x|+C\\ \int \frac{\mathrm{dx}}{ax+b}=\frac{1}{a}\ln |ax+b|+C\\ \int \frac{ax+b}{cx+d}\mathrm{dx}=\frac{a}{c}x+\frac{bc-ad}{c^2}\ln |cx+d|+C\\ \int \frac{\mathrm{dx}}{x^2+a^2}=\frac{1}{a}{\tan }^{-1}\frac{x}{a}+C\\ \int \frac{\mathrm{dx}}{x^2-a^2}=\frac{1}{2a}\ln |\frac{x-a}{x+a}|+C\\ \int \frac{\mathrm{dx}}{a^2-x^2}=\frac{1}{2a}\ln |\frac{a+x}{a-x}|+C\\ \int \frac{\mathrm{dx}}{(x+a)(x+b)}=\frac{1}{a-b}\ln |\frac{b+x}{a+x}|+C(a\ne b)\\ \int \frac{x\mathrm{dx}}{(x+a)(x+b)}=\frac{1}{a-b}(a\ln |x+a|-b\ln |x+b|)+C(a\ne b)\\ \int \frac{\mathrm{dx}}{x^2+a^2}=\frac{1}{a}\arctan \frac{x}{a}+C\\ \int \frac{x\mathrm{dx}}{x^2-a^2}=\frac{1}{2}\ln |x^2-a^2|+C\\ \int \frac{x\mathrm{dx}}{x^2+a^2}=\frac{1}{2}\ln |x^2+a^2|+C\\ \int \frac{\mathrm{dx}}{{(x^2+a^2)}^2}=\frac{1}{2a^2}\frac{x}{x^2+a^2}+\frac{1}{2a^3}\arctan \frac{x}{a}+C\\ \int \frac{x\mathrm{dx}}{{(x^2+a^2)}^2}=-\frac{1}{2}\frac{1}{x^2+a^2}+C\\ \int \frac{\mathrm{dx}}{(x^2+a^2)(x+b)}=\frac{1}{a^2+b^2}(\ln \frac{|x+b|}{\sqrt{x^2+a^2}}+\frac{b}{a}\arctan \frac{x}{a})+C\\ \int \frac{x\mathrm{dx}}{(x^2+a^2)(x+b)}=\frac{1}{a^2+b^2}(\arctan \frac{x}{a}-b\ln \frac{|x+b|}{\sqrt{x^2+a^2}})+C\end{array}$

8. Coordinate transformation

   $\begin{array}{l}\int F(ax+b)\mathrm{dx}=\frac{1}{a}\int F(u)\mathrm{du}(u=ax+b)\\ \int F(\sqrt{ax+b})\mathrm{dx}=\frac{2}{a}\int uF(u)\mathrm{du}(u=\sqrt{ax+b})\\ \int F(\sqrt[n]{ax+b})\mathrm{dx}=\frac{n}{a}\int u^{n-1}F(u)\mathrm{du}(u=\sqrt[n]{ax+b})\\ \int F(\sqrt{a^2-x^2})\mathrm{dx}=a\int F(a\cos u)\cos u\mathrm{du}(x=a\sin \mathrm{u)}\\ \int F(\sqrt{a^2+x^2})\mathrm{dx}=a\int F(a\sec u){\sec }^{2}u\mathrm{du}(x=a\tan \mathrm{u)}\\ \int F(\sqrt{x^2-a^2})\mathrm{dx}=a\int F(a\tan u)\sec u\tan u\mathrm{du}(x=a\sec \mathrm{u)}\\ \int F(e^{ax})\mathrm{dx}=\frac{1}{a}\int \frac{F(u)}{u}\mathrm{du}(u=e^{ax})\\ \int F(\ln x)\mathrm{dx}=\int F(u)e^u\mathrm{du}(u=\ln x)\end{array}$

9. Definite integral

   $\begin{array}{l}\underset{a}{\overset{b}{\int }}f(x)=F(x)\underset{a}{\overset{b}{|}}=F(b)-F(a)(F^{\prime }(x)=f(x))\\ \underset{a}{\overset{b}{\int }}\{f(x)\pm g(x)\}\mathrm{dx}=\underset{a}{\overset{b}{\int }}f(x)\mathrm{dx}\pm \underset{a}{\overset{b}{\int }}g(x)\mathrm{dx}\\ \underset{a}{\overset{b}{\int }}Cf(x)\mathrm{dx}=C\underset{a}{\overset{b}{\int }}f(x)\mathrm{dx}\\ \underset{a}{\overset{b}{\int }}f(x)\mathrm{dx}=-\underset{b}{\overset{a}{\int }}f(x)\mathrm{dx}\\ \underset{a}{\overset{a}{\int }}f(x)\mathrm{dx}=0\\ \underset{a}{\overset{b}{\int }}f(x)\mathrm{dx}=\underset{a}{\overset{c}{\int }}f(x)\mathrm{dx}+\underset{c}{\overset{b}{\int }}f(x)\mathrm{dx}\\ \underset{a}{\overset{b}{\int }}f(x)\mathrm{dx}=(b-a)f(c)(a\le c\le b)\\ \underset{a}{\overset{b}{\int }}f(x)g(x)\mathrm{dx}=f(c)\underset{a}{\overset{b}{\int }}g(x)\mathrm{dx}(a\le c\le b,g(x)\ge 0)\end{array}$