Sách Đại Số/Phép Toán Đại Số/Tích Phân/Tích phân bất định

Tích phân bất định

Tích phân bất định của một hàm số toán có công thức tổng quát

$\underset{}{\overset{}{\int }}f(x)dx=F(x)+C$

Phép toán tích phân bất định là một phép toán cao cấp tìm diện tích dưới hình của hàm số f(x)

$\underset{}{\overset{}{\int }}f(x)dx$ = Diện tích dưới hình

Công thức

Tích Phân Hàm Số Thường

	Integral	Value	Remarks
1	$\int cdx$	$cx+C$
2	$\int x^{n}dx$	$\frac{x^{n+1}}{n+1}+C$	$n\ne -1$
3	$\int \frac{1}{x}dx$	$\ln \left|x\right|+C$
4	$\int \frac{1}{a^2+x^2}dx$	$\frac{1}{a}\arctan \frac{x}{a}+C$
5	$\int \frac{1}{\sqrt{a^2-x^2}}dx$	$\arcsin \frac{x}{a}+C$
6	$\int \frac{-1}{\sqrt{a^2-x^2}}dx$	$\arccos \frac{x}{a}+C$
7	$\int \frac{1}{x\sqrt{x^2-a^2}}dx$	$\frac{1}{a}\text{arcsec}\frac{|x|}{a}+C$
8	$\int \ln xdx$	$x\ln x-x+C$
9	$\int {\log }_{b}xdx$	$x{\log }_{b}x-x{\log }_{b}e+C$
10	$\int e^{x}dx$	$e^x+C$
11	$\int a^{x}dx$	$\frac{a^x}{\ln a}+C$
12	$\int \sin xdx$	$-\cos x+C$
13	$\int \cos xdx$	$\sin x+C$
14	$\int \tan xdx$	$-\ln |\cos x|+C$
15	$\int \cot xdx$	$\ln |\sin x|+C$
16	$\int \sec xdx$	$\ln |\sec x+\tan x|+C$
17	$\int \csc xdx$	$-\ln |\csc x+\cot x|+C$
18	$\int {\sec }^{2}xdx$	$\tan x+C$
19	$\int {\csc }^{2}xdx$	$-\cot x+C$
20	$\int \sec x\tan xdx$	$\sec x+C$
21	$\int \csc x\cot xdx$	$-\csc x+C$
22	$\int {\sin }^{2}xdx$	$\frac{1}{2}(x-\sin x\cos x)+C$
23	$\int {\cos }^{2}xdx$	$\frac{1}{2}(x+\sin x\cos x)+C$
24	$\int {\sin }^{n}xdx$	$-\frac{{\sin }^{n-1}x\cos x}{n}+\frac{n-1}{n}\int {\sin }^{n-2}xdx$
25	$\int {\cos }^{n}xdx$	$-\frac{{\cos }^{n-1}x\sin x}{n}+\frac{n-1}{n}\int {\cos }^{n-2}xdx$
26	$\int \arctan xdx$	$x\arctan x-\frac{1}{2}\ln |1+x^2|+C$
27	$\int \sinh xdx$	$\cosh x+C$
28	$\int \cosh xdx$	$\sinh x+C$
29	$\int \tanh xdx$	$\ln |\cosh x|+C$
30	$\int \text{csch}xdx$	$\ln |\tanh \frac{x}{2}|+C$
31	$\int \text{sech}xdx$	$\arctan (\sinh x)+C$
32	$\int \coth xdx$	$\ln |\sinh x|+C$

Tích Phân Hàm Số Hyperboly

Dưới đây là danh sách tích phân với hàm hypebolic.

$\int \sinh cxdx=\frac{1}{c}\cosh cx$

$\int \cosh cxdx=\frac{1}{c}\sinh cx$

$\int {\sinh }^{2}cxdx=\frac{1}{4c}\sinh 2cx-\frac{x}{2}$

$\int {\cosh }^{2}cxdx=\frac{1}{4c}\sinh 2cx+\frac{x}{2}$

$\int {\sinh }^{n}cxdx=\frac{1}{cn}{\sinh }^{n-1}cx\cosh cx-\frac{n-1}{n}\int {\sinh }^{n-2}cxdx\text{( }n>0\text{)}$

hay: $\int {\sinh }^{n}cxdx=\frac{1}{c(n+1)}{\sinh }^{n+1}cx\cosh cx-\frac{n+2}{n+1}\int {\sinh }^{n+2}cxdx\text{( }n<0\text{, }n\ne -1\text{)}$

$\int {\cosh }^{n}cxdx=\frac{1}{cn}\sinh cx{\cosh }^{n-1}cx+\frac{n-1}{n}\int {\cosh }^{n-2}cxdx\text{( }n>0\text{)}$

hay: $\int {\cosh }^{n}cxdx=-\frac{1}{c(n+1)}\sinh cx{\cosh }^{n+1}cx-\frac{n+2}{n+1}\int {\cosh }^{n+2}cxdx\text{( }n<0\text{, }n\ne -1\text{)}$

$\int \frac{dx}{\sinh cx}=\frac{1}{c}\ln |\tanh \frac{cx}{2}|$

hay: $\int \frac{dx}{\sinh cx}=\frac{1}{c}\ln \left|\frac{\cosh cx-1}{\sinh cx}\right|$

hay: $\int \frac{dx}{\sinh cx}=\frac{1}{c}\ln \left|\frac{\sinh cx}{\cosh cx+1}\right|$

hay: $\int \frac{dx}{\sinh cx}=\frac{1}{c}\ln \left|\frac{\cosh cx-1}{\cosh cx+1}\right|$

$\int \frac{dx}{\cosh cx}=\frac{2}{c}\arctan e^{cx}$

$\int \frac{dx}{{\sinh }^{n}cx}=\frac{\cosh cx}{c(n-1){\sinh }^{n-1}cx}-\frac{n-2}{n-1}\int \frac{dx}{{\sinh }^{n-2}cx}\text{( }n\ne 1\text{)}$

$\int \frac{dx}{{\cosh }^{n}cx}=\frac{\sinh cx}{c(n-1){\cosh }^{n-1}cx}+\frac{n-2}{n-1}\int \frac{dx}{{\cosh }^{n-2}cx}\text{( }n\ne 1\text{)}$

$\int \frac{{\cosh }^{n}cx}{{\sinh }^{m}cx}dx=\frac{{\cosh }^{n-1}cx}{c(n-m){\sinh }^{m-1}cx}+\frac{n-1}{n-m}\int \frac{{\cosh }^{n-2}cx}{{\sinh }^{m}cx}dx\text{( }m\ne n\text{)}$

hay: $\int \frac{{\cosh }^{n}cx}{{\sinh }^{m}cx}dx=-\frac{{\cosh }^{n+1}cx}{c(m-1){\sinh }^{m-1}cx}+\frac{n-m+2}{m-1}\int \frac{{\cosh }^{n}cx}{{\sinh }^{m-2}cx}dx\text{( }m\ne 1\text{)}$

hay: $\int \frac{{\cosh }^{n}cx}{{\sinh }^{m}cx}dx=-\frac{{\cosh }^{n-1}cx}{c(m-1){\sinh }^{m-1}cx}+\frac{n-1}{m-1}\int \frac{{\cosh }^{n-2}cx}{{\sinh }^{m-2}cx}dx\text{( }m\ne 1\text{)}$

$\int \frac{{\sinh }^{m}cx}{{\cosh }^{n}cx}dx=\frac{{\sinh }^{m-1}cx}{c(m-n){\cosh }^{n-1}cx}+\frac{m-1}{m-n}\int \frac{{\sinh }^{m-2}cx}{{\cosh }^{n}cx}dx\text{( }m\ne n\text{)}$

hay: $\int \frac{{\sinh }^{m}cx}{{\cosh }^{n}cx}dx=\frac{{\sinh }^{m+1}cx}{c(n-1){\cosh }^{n-1}cx}+\frac{m-n+2}{n-1}\int \frac{{\sinh }^{m}cx}{{\cosh }^{n-2}cx}dx\text{( }n\ne 1\text{)}$

hay: $\int \frac{{\sinh }^{m}cx}{{\cosh }^{n}cx}dx=-\frac{{\sinh }^{m-1}cx}{c(n-1){\cosh }^{n-1}cx}+\frac{m-1}{n-1}\int \frac{{\sinh }^{m-2}cx}{{\cosh }^{n-2}cx}dx\text{( }n\ne 1\text{)}$

$\int x\sinh cxdx=\frac{1}{c}x\cosh cx-\frac{1}{c^2}\sinh cx$

$\int x\cosh cxdx=\frac{1}{c}x\sinh cx-\frac{1}{c^2}\cosh cx$

$\int \tanh cxdx=\frac{1}{c}\ln |\cosh cx|$

$\int \coth cxdx=\frac{1}{c}\ln |\sinh cx|$

$\int {\tanh }^{n}cxdx=-\frac{1}{c(n-1)}{\tanh }^{n-1}cx+\int {\tanh }^{n-2}cxdx\text{( }n\ne 1\text{)}$

$\int {\coth }^{n}cxdx=-\frac{1}{c(n-1)}{\coth }^{n-1}cx+\int {\coth }^{n-2}cxdx\text{( }n\ne 1\text{)}$

$\int \sinh bx\sinh cxdx=\frac{1}{b^2-c^2}(b\sinh cx\cosh bx-c\cosh cx\sinh bx)\text{( }{b}^2\ne c^2\text{)}$

$\int \cosh bx\cosh cxdx=\frac{1}{b^2-c^2}(b\sinh bx\cosh cx-c\sinh cx\cosh bx)\text{( }{b}^2\ne c^2\text{)}$

$\int \cosh bx\sinh cxdx=\frac{1}{b^2-c^2}(b\sinh bx\sinh cx-c\cosh bx\cosh cx)\text{( }{b}^2\ne c^2\text{)}$

$\int \sinh (ax+b)\sin (cx+d)dx=\frac{a}{a^2+c^2}\cosh (ax+b)\sin (cx+d)-\frac{c}{a^2+c^2}\sinh (ax+b)\cos (cx+d)$

$\int \sinh (ax+b)\cos (cx+d)dx=\frac{a}{a^2+c^2}\cosh (ax+b)\cos (cx+d)+\frac{c}{a^2+c^2}\sinh (ax+b)\sin (cx+d)$

$\int \cosh (ax+b)\sin (cx+d)dx=\frac{a}{a^2+c^2}\sinh (ax+b)\sin (cx+d)-\frac{c}{a^2+c^2}\cosh (ax+b)\cos (cx+d)$

$\int \cosh (ax+b)\cos (cx+d)dx=\frac{a}{a^2+c^2}\sinh (ax+b)\cos (cx+d)+\frac{c}{a^2+c^2}\cosh (ax+b)\sin (cx+d)$

Tích Phân Hàm Số Hyperboly Ngược

Dưới đây là danh sách các tích phân với hàm hypebolic ngược.

$\int \mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\frac{x}{c}dx=x\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\frac{x}{c}-\sqrt{x^2+c^2}$

$\int \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\frac{x}{c}dx=x\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\frac{x}{c}-\sqrt{x^2-c^2}$

$\int \mathrm{a}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\frac{x}{c}dx=x\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\frac{x}{c}+\frac{c}{2}\ln |c^2-x^2|\text{( }|x|<|c|\text{)}$

$\int \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\frac{x}{c}dx=x\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\frac{x}{c}+\frac{c}{2}\ln |x^2-c^2|\text{( }|x|>|c|\text{)}$

$\int \mathrm{a}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\frac{x}{c}dx=x\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\frac{x}{c}-c\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\frac{x\sqrt{\frac{c-x}{c+x}}}{x-c}\text{( }x\in (0,c)\text{)}$

$\int \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}\frac{x}{c}dx=x\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}\frac{x}{c}+c\ln \frac{x+\sqrt{x^2+c^2}}{c}\text{( }x\in (0,c)\text{)}$

Tích phân hàm số Logarit

Dưới đây là danh sách tích phân với hàm lôgarít.

Chú ý: bài này quy ước x>0.

$\int \ln cxdx=x\ln cx-x$

• $\int (\ln x{)}^{2}dx=x(\ln x{)}^2-2x\ln x+2x$

• $\int (\ln cx{)}^{n}dx=x(\ln cx{)}^n-n\int (\ln cx{)}^{n-1}dx$

• $\int \frac{dx}{\ln x}=\ln |\ln x|+\ln x+\sum _{i=2}^{\mathrm{\infty }}\frac{(\ln x{)}^i}{i\cdot i!}$

• $\int \frac{dx}{(\ln x{)}^n}=-\frac{x}{(n-1)(\ln x{)}^{n-1}}+\frac{1}{n-1}\int \frac{dx}{(\ln x{)}^{n-1}}\text{( }n\ne 1\text{)}$

• $\int x^m\ln xdx=x^{m+1}(\frac{\ln x}{m+1}-\frac{1}{(m+1{)}^2})\text{( }m\ne -1\text{)}$

• $\int x^m(\ln x{)}^{n}dx=\frac{x^{m+1}(\ln x{)}^n}{m+1}-\frac{n}{m+1}\int x^m(\ln x{)}^{n-1}dx\text{( }m\ne -1\text{)}$

• $\int \frac{(\ln x{)}^{n}dx}{x}=\frac{(\ln x{)}^{n+1}}{n+1}\text{( }n\ne -1\text{)}$

• $\int \frac{\ln xdx}{x^m}=-\frac{\ln x}{(m-1)x^{m-1}}-\frac{1}{(m-1{)}^2{x}^{m-1}}\text{( }m\ne 1\text{)}$

• $\int \frac{(\ln x{)}^{n}dx}{x^m}=-\frac{(\ln x{)}^n}{(m-1)x^{m-1}}+\frac{n}{m-1}\int \frac{(\ln x{)}^{n-1}dx}{x^m}\text{( }m\ne 1\text{)}$

• $\int \frac{x^{m}dx}{(\ln x{)}^n}=-\frac{x^{m+1}}{(n-1)(\ln x{)}^{n-1}}+\frac{m+1}{n-1}\int \frac{x^{m}dx}{(\ln x{)}^{n-1}}\text{( }n\ne 1\text{)}$

• $\int \frac{dx}{x\ln x}=\ln |\ln x|$

• $\int \frac{dx}{x^n\ln x}=\ln |\ln x|+\sum _{i=1}^{\mathrm{\infty }}(-1{)}^i\frac{(n-1{)}^i(\ln x{)}^i}{i\cdot i!}$

• $\int \frac{dx}{x(\ln x{)}^n}=-\frac{1}{(n-1)(\ln x{)}^{n-1}}\text{( }n\ne 1\text{)}$

• $\int \sin (\ln x)dx=\frac{x}{2}(\sin (\ln x)-\cos (\ln x))$

• $\int \cos (\ln x)dx=\frac{x}{2}(\sin (\ln x)+\cos (\ln x))$

Tích phân hàm số mũ

Dưới đây là danh sách các tích phân với hàm mũ.

$\int e^{cx}dx=\frac{1}{c}{e}^{cx}$

$\int a^{cx}dx=\frac{1}{c\ln a}{a}^{cx}\text{( }a>0,a\ne 1\text{)}$

$\int x{e}^{cx}dx=\frac{e^{cx}}{c^2}(cx-1)$

$\int x^2{e}^{cx}dx=e^{cx}(\frac{x^2}{c}-\frac{2x}{c^2}+\frac{2}{c^3})$

$\int x^n{e}^{cx}dx=\frac{1}{c}{x}^n{e}^{cx}-\frac{n}{c}\int x^{n-1}{e}^{cx}dx$

$\int \frac{e^{cx}dx}{x}=\ln |x|+\sum _{i=1}^{\mathrm{\infty }}\frac{(cx{)}^i}{i\cdot i!}$

$\int \frac{e^{cx}dx}{x^n}=\frac{1}{n-1}(-\frac{e^{cx}}{x^{n-1}}+c\int \frac{e^{cx}}{x^{n-1}}dx)\text{( }n\ne 1\text{)}$

$\int e^{cx}\ln xdx=\frac{1}{c}{e}^{cx}\ln |x|-\mathrm{Ei}(cx)$

$\int e^{cx}\sin bxdx=\frac{e^{cx}}{c^2+b^2}(c\sin bx-b\cos bx)$

$\int e^{cx}\cos bxdx=\frac{e^{cx}}{c^2+b^2}(c\cos bx+b\sin bx)$

$\int e^{cx}{\sin }^{n}xdx=\frac{e^{cx}{\sin }^{n-1}x}{c^2+n^2}(c\sin x-n\cos x)+\frac{n(n-1)}{c^2+n^2}\int e^{cx}{\sin }^{n-2}xdx$

$\int e^{cx}{\cos }^{n}xdx=\frac{e^{cx}{\cos }^{n-1}x}{c^2+n^2}(c\cos x+n\sin x)+\frac{n(n-1)}{c^2+n^2}\int e^{cx}{\cos }^{n-2}xdx$

$\int x{e}^{c{x}^2}dx=\frac{1}{2c}{e}^{c{x}^2}$

$\int \frac{1}{\sigma \sqrt{2\pi }}{e}^{-(x-\mu {)}^2/2{\sigma }^2}dx=\frac{1}{2\sigma }(1+\text{erf}\frac{x-\mu }{\sigma \sqrt{2}})$

$\int e^{x^2}dx=e^{x^2}\left(\sum _{j=0}^{n-1}{c}_{2j}\frac{1}{x^{2j+1}}\right)+(2n-1)c_{2n-2}\int \frac{e^{x^2}}{x^{2n}}dx(n>0),$

với $c_{2j}=\frac{1\cdot 3\cdot 5\cdots (2j-1)}{2^{j+1}}=\frac{2j!}{j!2^{2j+1}}.$

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-a{x}^2}dx=\sqrt{\frac{\pi }{a}}$

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^{2n}{e}^{-x^2/a^2}dx=\sqrt{\pi }\frac{(2n)!}{n!}{\left(\frac{a}{2}\right)}^{2n+1}$

Tích phân hàm số lượng giác

Tích phân hàm số sine

$\int \sin axdx=-\frac{1}{a}\cos ax+C$

$\int {\sin }^{2}axdx=\frac{x}{2}-\frac{1}{4a}\sin 2ax+C=\frac{x}{2}-\frac{1}{2a}\sin ax\cos ax+C$

$\int x{\sin }^{2}axdx=\frac{x^2}{4}-\frac{x}{4a}\sin 2ax-\frac{1}{8a^2}\cos 2ax+C$

$\int x^2{\sin }^{2}axdx=\frac{x^3}{6}-(\frac{x^2}{4a}-\frac{1}{8a^3})\sin 2ax-\frac{x}{4a^2}\cos 2ax+C$

$\int \sin b_{1}x\sin b_{2}xdx=\frac{\sin ((b_1-b_2)x)}{2(b_1-b_2)}-\frac{\sin ((b_1+b_2)x)}{2(b_1+b_2)}+C\text{(for }|b_1|\ne |b_2|\text{)}$

$\int {\sin }^{n}axdx=-\frac{{\sin }^{n-1}ax\cos ax}{na}+\frac{n-1}{n}\int {\sin }^{n-2}axdx\text{(for }n>0\text{)}$

$\int \frac{dx}{\sin ax}=\frac{1}{a}\ln |\tan \frac{ax}{2}|+C$

$\int \frac{dx}{{\sin }^{n}ax}=\frac{\cos ax}{a(1-n){\sin }^{n-1}ax}+\frac{n-2}{n-1}\int \frac{dx}{{\sin }^{n-2}ax}\text{(for }n>1\text{)}$

$\int x\sin axdx=\frac{\sin ax}{a^2}-\frac{x\cos ax}{a}+C$

$\int x^n\sin axdx=-\frac{x^n}{a}\cos ax+\frac{n}{a}\int x^{n-1}\cos axdx\text{(for }n>0\text{)}$

$\underset{\frac{-a}{2}}{\overset{\frac{a}{2}}{\int }}{x}^2{\sin }^2\frac{n\pi x}{a}dx=\frac{a^3(n^2{\pi }^2-6)}{24n^2{\pi }^2}\text{(for }n=2,4,6...\text{)}$

$\int \frac{\sin ax}{x}dx=\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n\frac{(ax{)}^{2n+1}}{(2n+1)\cdot (2n+1)!}+C$

$\int \frac{\sin ax}{x^n}dx=-\frac{\sin ax}{(n-1)x^{n-1}}+\frac{a}{n-1}\int \frac{\cos ax}{x^{n-1}}dx$

$\int \frac{dx}{1\pm \sin ax}=\frac{1}{a}\tan (\frac{ax}{2}\mp \frac{\pi }{4})+C$

$\int \frac{xdx}{1+\sin ax}=\frac{x}{a}\tan (\frac{ax}{2}-\frac{\pi }{4})+\frac{2}{a^2}\ln |\cos (\frac{ax}{2}-\frac{\pi }{4})|+C$

$\int \frac{xdx}{1-\sin ax}=\frac{x}{a}\cot (\frac{\pi }{4}-\frac{ax}{2})+\frac{2}{a^2}\ln |\sin (\frac{\pi }{4}-\frac{ax}{2})|+C$

$\int \frac{\sin axdx}{1\pm \sin ax}=\pm x+\frac{1}{a}\tan (\frac{\pi }{4}\mp \frac{ax}{2})+C$

• Tích phân bất định cosine

$\int \cos axdx=\frac{1}{a}\sin ax+C$

$\int {\cos }^{n}axdx=\frac{{\cos }^{n-1}ax\sin ax}{na}+\frac{n-1}{n}\int {\cos }^{n-2}axdx\text{(for }n>0\text{)}$

$\int x\cos axdx=\frac{\cos ax}{a^2}+\frac{x\sin ax}{a}+C$

$\int {\cos }^{2}axdx=\frac{x}{2}+\frac{1}{4a}\sin 2ax+C=\frac{x}{2}+\frac{1}{2a}\sin ax\cos ax+C$

$\int x^2{\cos }^{2}axdx=\frac{x^3}{6}+(\frac{x^2}{4a}-\frac{1}{8a^3})\sin 2ax+\frac{x}{4a^2}\cos 2ax+C$

$\int x^n\cos axdx=\frac{x^n\sin ax}{a}-\frac{n}{a}\int x^{n-1}\sin axdx$

$\int \frac{\cos ax}{x}dx=\ln |ax|+\sum _{k=1}^{\mathrm{\infty }}(-1{)}^k\frac{(ax{)}^{2k}}{2k\cdot (2k)!}+C$

$\int \frac{\cos ax}{x^n}dx=-\frac{\cos ax}{(n-1)x^{n-1}}-\frac{a}{n-1}\int \frac{\sin ax}{x^{n-1}}dx\text{(for }n\ne 1\text{)}$

$\int \frac{dx}{\cos ax}=\frac{1}{a}\ln |\tan (\frac{ax}{2}+\frac{\pi }{4})|+C$

$\int \frac{dx}{{\cos }^{n}ax}=\frac{\sin ax}{a(n-1){\cos }^{n-1}ax}+\frac{n-2}{n-1}\int \frac{dx}{{\cos }^{n-2}ax}\text{(for }n>1\text{)}$

$\int \frac{dx}{1+\cos ax}=\frac{1}{a}\tan \frac{ax}{2}+C$

$\int \frac{dx}{1-\cos ax}=-\frac{1}{a}\cot \frac{ax}{2}+C$

$\int \frac{xdx}{1+\cos ax}=\frac{x}{a}\tan \frac{ax}{2}+\frac{2}{a^2}\ln |\cos \frac{ax}{2}|+C$

$\int \frac{xdx}{1-\cos ax}=-\frac{x}{a}\cot \frac{ax}{2}+\frac{2}{a^2}\ln |\sin \frac{ax}{2}|+C$

$\int \frac{\cos axdx}{1+\cos ax}=x-\frac{1}{a}\tan \frac{ax}{2}+C$

$\int \frac{\cos axdx}{1-\cos ax}=-x-\frac{1}{a}\cot \frac{ax}{2}+C$

$\int \cos a_{1}x\cos a_{2}xdx=\frac{\sin (a_1-a_2)x}{2(a_1-a_2)}+\frac{\sin (a_1+a_2)x}{2(a_1+a_2)}+C\text{(for }|a_1|\ne |a_2|\text{)}$

• Tích phân bất địnhtangent

$\int \tan axdx=-\frac{1}{a}\ln |\cos ax|+C=\frac{1}{a}\ln |\sec ax|+C$

$\int {\tan }^{n}axdx=\frac{1}{a(n-1)}{\tan }^{n-1}ax-\int {\tan }^{n-2}axdx\text{(for }n\ne 1\text{)}$

$\int \frac{dx}{q\tan ax+p}=\frac{1}{p^2+q^2}(px+\frac{q}{a}\ln |q\sin ax+p\cos ax|)+C\text{(for }{p}^2+q^2\ne 0\text{)}$

$\int \frac{dx}{\tan ax}=\frac{1}{a}\ln |\sin ax|+C$

$\int \frac{dx}{\tan ax+1}=\frac{x}{2}+\frac{1}{2a}\ln |\sin ax+\cos ax|+C$

$\int \frac{dx}{\tan ax-1}=-\frac{x}{2}+\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

$\int \frac{\tan axdx}{\tan ax+1}=\frac{x}{2}-\frac{1}{2a}\ln |\sin ax+\cos ax|+C$

$\int \frac{\tan axdx}{\tan ax-1}=\frac{x}{2}+\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

• Tích phân bất địnhonly secant

$\int \sec axdx=\frac{1}{a}\ln |\sec ax+\tan ax|+C$

$\int {\sec }^{n}axdx=\frac{{\sec }^{n-1}ax\sin ax}{a(n-1)}+\frac{n-2}{n-1}\int {\sec }^{n-2}axdx\text{ (for }n\ne 1\text{)}$

$\int {\sec }^{n}xdx=\frac{{\sec }^{n-2}x\tan x}{n-1}+\frac{n-2}{n-1}\int {\sec }^{n-2}xdx$

$\int \frac{dx}{\sec x+1}=x-\tan \frac{x}{2}+C$

• Tích phân bất định cosecant

$\int \csc axdx=\frac{1}{a}\ln |\csc ax-\cot ax|+C$

$\int {\csc }^{2}xdx=-\cot x+C$

$\int {\csc }^{n}axdx=-\frac{{\csc }^{n-1}ax\cos ax}{a(n-1)}+\frac{n-2}{n-1}\int {\csc }^{n-2}axdx\text{ (for }n\ne 1\text{)}$

• Tích phân bất định cotangent

$\int \cot axdx=\frac{1}{a}\ln |\sin ax|+C$

$\int {\cot }^{n}axdx=-\frac{1}{a(n-1)}{\cot }^{n-1}ax-\int {\cot }^{n-2}axdx\text{(for }n\ne 1\text{)}$

$\int \frac{dx}{1+\cot ax}=\int \frac{\tan axdx}{\tan ax+1}$

$\int \frac{dx}{1-\cot ax}=\int \frac{\tan axdx}{\tan ax-1}$

• Tích phân bất địnhsine and cosine

$\int \frac{dx}{\cos ax\pm \sin ax}=\frac{1}{a\sqrt{2}}\ln |\tan (\frac{ax}{2}\pm \frac{\pi }{8})|+C$

$\int \frac{dx}{(\cos ax\pm \sin ax{)}^2}=\frac{1}{2a}\tan (ax\mp \frac{\pi }{4})+C$

$\int \frac{dx}{(\cos x+\sin x{)}^n}=\frac{1}{n-1}(\frac{\sin x-\cos x}{(\cos x+\sin x{)}^{n-1}}-2(n-2)\int \frac{dx}{(\cos x+\sin x{)}^{n-2}})$

$\int \frac{\cos axdx}{\cos ax+\sin ax}=\frac{x}{2}+\frac{1}{2a}\ln |\sin ax+\cos ax|+C$

$\int \frac{\cos axdx}{\cos ax-\sin ax}=\frac{x}{2}-\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

$\int \frac{\sin axdx}{\cos ax+\sin ax}=\frac{x}{2}-\frac{1}{2a}\ln |\sin ax+\cos ax|+C$

$\int \frac{\sin axdx}{\cos ax-\sin ax}=-\frac{x}{2}-\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

$\int \frac{\cos axdx}{\sin ax(1+\cos ax)}=-\frac{1}{4a}{\tan }^2\frac{ax}{2}+\frac{1}{2a}\ln |\tan \frac{ax}{2}|+C$

$\int \frac{\cos axdx}{\sin ax(1+-\cos ax)}=-\frac{1}{4a}{\cot }^2\frac{ax}{2}-\frac{1}{2a}\ln |\tan \frac{ax}{2}|+C$

$\int \frac{\sin axdx}{\cos ax(1+\sin ax)}=\frac{1}{4a}{\cot }^2(\frac{ax}{2}+\frac{\pi }{4})+\frac{1}{2a}\ln |\tan (\frac{ax}{2}+\frac{\pi }{4})|+C$

$\int \frac{\sin axdx}{\cos ax(1-\sin ax)}=\frac{1}{4a}{\tan }^2(\frac{ax}{2}+\frac{\pi }{4})-\frac{1}{2a}\ln |\tan (\frac{ax}{2}+\frac{\pi }{4})|+C$

$\int \sin ax\cos axdx=\frac{1}{2a}{\sin }^{2}ax+C$

$\int \sin a_{1}x\cos a_{2}xdx=-\frac{\cos (a_1+a_2)x}{2(a_1+a_2)}-\frac{\cos (a_1-a_2)x}{2(a_1-a_2)}+C\text{(for }|a_1|\ne |a_2|\text{)}$

$\int {\sin }^{n}ax\cos axdx=\frac{1}{a(n+1)}{\sin }^{n+1}ax+C\text{(for }n\ne -1\text{)}$

$\int \sin ax{\cos }^{n}axdx=-\frac{1}{a(n+1)}{\cos }^{n+1}ax+C\text{(for }n\ne -1\text{)}$

$\int {\sin }^{n}ax{\cos }^{m}axdx=-\frac{{\sin }^{n-1}ax{\cos }^{m+1}ax}{a(n+m)}+\frac{n-1}{n+m}\int {\sin }^{n-2}ax{\cos }^{m}axdx\text{(for }m,n>0\text{)}$

also: $\int {\sin }^{n}ax{\cos }^{m}axdx=\frac{{\sin }^{n+1}ax{\cos }^{m-1}ax}{a(n+m)}+\frac{m-1}{n+m}\int {\sin }^{n}ax{\cos }^{m-2}axdx\text{(for }m,n>0\text{)}$

$\int \frac{dx}{\sin ax\cos ax}=\frac{1}{a}\ln |\tan ax|+C$

$\int \frac{dx}{\sin ax{\cos }^{n}ax}=\frac{1}{a(n-1){\cos }^{n-1}ax}+\int \frac{dx}{\sin ax{\cos }^{n-2}ax}\text{(for }n\ne 1\text{)}$

$\int \frac{dx}{{\sin }^{n}ax\cos ax}=-\frac{1}{a(n-1){\sin }^{n-1}ax}+\int \frac{dx}{{\sin }^{n-2}ax\cos ax}\text{(for }n\ne 1\text{)}$

$\int \frac{\sin axdx}{{\cos }^{n}ax}=\frac{1}{a(n-1){\cos }^{n-1}ax}+C\text{(for }n\ne 1\text{)}$

$\int \frac{{\sin }^{2}axdx}{\cos ax}=-\frac{1}{a}\sin ax+\frac{1}{a}\ln |\tan (\frac{\pi }{4}+\frac{ax}{2})|+C$

$\int \frac{{\sin }^{2}axdx}{{\cos }^{n}ax}=\frac{\sin ax}{a(n-1){\cos }^{n-1}ax}-\frac{1}{n-1}\int \frac{dx}{{\cos }^{n-2}ax}\text{(for }n\ne 1\text{)}$

$\int \frac{{\sin }^{n}axdx}{\cos ax}=-\frac{{\sin }^{n-1}ax}{a(n-1)}+\int \frac{{\sin }^{n-2}axdx}{\cos ax}\text{(for }n\ne 1\text{)}$

$\int \frac{{\sin }^{n}axdx}{{\cos }^{m}ax}=\frac{{\sin }^{n+1}ax}{a(m-1){\cos }^{m-1}ax}-\frac{n-m+2}{m-1}\int \frac{{\sin }^{n}axdx}{{\cos }^{m-2}ax}\text{(for }m\ne 1\text{)}$

also: $\int \frac{{\sin }^{n}axdx}{{\cos }^{m}ax}=-\frac{{\sin }^{n-1}ax}{a(n-m){\cos }^{m-1}ax}+\frac{n-1}{n-m}\int \frac{{\sin }^{n-2}axdx}{{\cos }^{m}ax}\text{(for }m\ne n\text{)}$

also: $\int \frac{{\sin }^{n}axdx}{{\cos }^{m}ax}=\frac{{\sin }^{n-1}ax}{a(m-1){\cos }^{m-1}ax}-\frac{n-1}{m-1}\int \frac{{\sin }^{n-2}axdx}{{\cos }^{m-2}ax}\text{(for }m\ne 1\text{)}$

$\int \frac{\cos axdx}{{\sin }^{n}ax}=-\frac{1}{a(n-1){\sin }^{n-1}ax}+C\text{(for }n\ne 1\text{)}$

$\int \frac{{\cos }^{2}axdx}{\sin ax}=\frac{1}{a}(\cos ax+\ln |\tan \frac{ax}{2}|)+C$

$\int \frac{{\cos }^{2}axdx}{{\sin }^{n}ax}=-\frac{1}{n-1}(\frac{\cos ax}{a{\sin }^{n-1}ax)}+\int \frac{dx}{{\sin }^{n-2}ax})\text{(for }n\ne 1\text{)}$

$\int \frac{{\cos }^{n}axdx}{{\sin }^{m}ax}=-\frac{{\cos }^{n+1}ax}{a(m-1){\sin }^{m-1}ax}-\frac{n-m-2}{m-1}\int \frac{{\cos }^{n}axdx}{{\sin }^{m-2}ax}\text{(for }m\ne 1\text{)}$

also: $\int \frac{{\cos }^{n}axdx}{{\sin }^{m}ax}=\frac{{\cos }^{n-1}ax}{a(n-m){\sin }^{m-1}ax}+\frac{n-1}{n-m}\int \frac{{\cos }^{n-2}axdx}{{\sin }^{m}ax}\text{(for }m\ne n\text{)}$

also: $\int \frac{{\cos }^{n}axdx}{{\sin }^{m}ax}=-\frac{{\cos }^{n-1}ax}{a(m-1){\sin }^{m-1}ax}-\frac{n-1}{m-1}\int \frac{{\cos }^{n-2}axdx}{{\sin }^{m-2}ax}\text{(for }m\ne 1\text{)}$

• Tích phân bất định sine and tangent

$\int \sin ax\tan axdx=\frac{1}{a}(\ln |\sec ax+\tan ax|-\sin ax)+C$

$\int \frac{{\tan }^{n}axdx}{{\sin }^{2}ax}=\frac{1}{a(n-1)}{\tan }^{n-1}(ax)+C\text{(for }n\ne 1\text{)}$

• Tích phân bất định cosine and tangent

$\int \frac{{\tan }^{n}axdx}{{\cos }^{2}ax}=\frac{1}{a(n+1)}{\tan }^{n+1}ax+C\text{(for }n\ne -1\text{)}$

• Tích phân bất định sine and cotangent

$\int \frac{{\cot }^{n}axdx}{{\sin }^{2}ax}=\frac{1}{a(n+1)}{\cot }^{n+1}ax+C\text{(for }n\ne -1\text{)}$

• Tích phân bất địnhcosine and cotangent

$\int \frac{{\cot }^{n}axdx}{{\cos }^{2}ax}=\frac{1}{a(1-n)}{\tan }^{1-n}ax+C\text{(for }n\ne 1\text{)}$

• Tích phân bất định

$\underset{-c}{\overset{c}{\int }}\sin xdx=0$

$\underset{-c}{\overset{c}{\int }}\cos xdx=2\underset{0}{\overset{c}{\int }}\cos xdx=2\underset{-c}{\overset{0}{\int }}\cos xdx=2\sin c$

$\underset{-c}{\overset{c}{\int }}\tan xdx=0$

$\underset{\frac{-a}{2}}{\overset{\frac{a}{2}}{\int }}{x}^2{\cos }^2\frac{n\pi x}{a}dx=\frac{a^3(n^2{\pi }^2-6)}{24n^2{\pi }^2}\text{(for }n=1,3,5...\text{)}$

Tích phân hàm lượng giác ngược

Dưới đây là danh sách các tích phân với hàm lượng giác ngược.

$\int \arcsin \frac{x}{c}dx=x\arcsin \frac{x}{c}+\sqrt{c^2-x^2}$

$\int x\arcsin \frac{x}{c}dx=(\frac{x^2}{2}-\frac{c^2}{4})\arcsin \frac{x}{c}+\frac{x}{4}\sqrt{c^2-x^2}$

$\int x^2\arcsin \frac{x}{c}dx=\frac{x^3}{3}\arcsin \frac{x}{c}+\frac{x^2+2c^2}{9}\sqrt{c^2-x^2}$

$\int x^n{\sin }^{-1}xdx=\frac{1}{n+1}(x^{n+1}{\sin }^{-1}x$

$+\frac{x^n\sqrt{1-x^2}-n{x}^{n-1}{\sin }^{-1}x}{n-1}+n\int x^{n-2}{\sin }^{-1}xdx)$

$\int \arccos \frac{x}{c}dx=x\arccos \frac{x}{c}-\sqrt{c^2-x^2}$

$\int x\arccos \frac{x}{c}dx=(\frac{x^2}{2}-\frac{c^2}{4})\arccos \frac{x}{c}-\frac{x}{4}\sqrt{c^2-x^2}$

$\int x^2\arccos \frac{x}{c}dx=\frac{x^3}{3}\arccos \frac{x}{c}-\frac{x^2+2c^2}{9}\sqrt{c^2-x^2}$

$\int \arctan \frac{x}{c}dx=x\arctan \frac{x}{c}-\frac{c}{2}\ln (c^2+x^2)$

$\int x\arctan \frac{x}{c}dx=\frac{c^2+x^2}{2}\arctan \frac{x}{c}-\frac{cx}{2}$

$\int x^2\arctan \frac{x}{c}dx=\frac{x^3}{3}\arctan \frac{x}{c}-\frac{c{x}^2}{6}+\frac{c^3}{6}\ln c^2+x^2$

$\int x^n\arctan \frac{x}{c}dx=\frac{x^{n+1}}{n+1}\arctan \frac{x}{c}-\frac{c}{n+1}\int \frac{x^{n+1}dx}{c^2+x^2}\text{( }n\ne 1\text{)}$

$\int \mathrm{arcsec}\frac{x}{c}dx=x\mathrm{arcsec}\frac{x}{c}+\frac{x}{c|x|}\ln |x\pm \sqrt{x^2-1}|$

$\int x\mathrm{arcsec}xdx=\frac{1}{2}(x^2\mathrm{arcsec}x-\sqrt{x^2-1})$

$\int x^n\mathrm{arcsec}xdx=\frac{1}{n+1}(x^{n+1}\mathrm{arcsec}x-\frac{1}{n}(x^{n-1}\sqrt{x^2-1}$

$+(1-n)(x^{n-1}\mathrm{arcsec}x+(1-n)\int x^{n-2}\mathrm{arcsec}xdx)))$

$\int \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}dx=x\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}+\frac{c}{2}\ln (c^2+x^2)$

$\int x\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}dx=\frac{c^2+x^2}{2}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}+\frac{cx}{2}$

$\int x^2\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}dx=\frac{x^3}{3}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}+\frac{c{x}^2}{6}-\frac{c^3}{6}\ln (c^2+x^2)$

$\int x^n\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}dx=\frac{x^{n+1}}{n+1}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}+\frac{c}{n+1}\int \frac{x^{n+1}dx}{c^2+x^2}\text{( }n\ne 1\text{)}$