工程表格/積分表

這是一個關於在這裡找到的恆等式的簡短摘要。

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