적분공식 총정리

각 공식의 증명은 번호(No.)를 클릭하여 링크를 참조한다.

 

No.	\(f(x)\)	\(\int f(x)dx\)	비고
1	1	x
2	\(x^\alpha\)	\(\frac{x^{\alpha+1}}{\alpha+1}\)	\(\alpha\ne0\)
3	\(1/x\)	\(\ln{x}\)
4	\(e^x\)	\(e^x\)
5	\(\cos{x}\)	\(\sin{x}\)
6	\(\sin{x}\)	\(-\cos{x}\)
7	\(\sec^2{x}\)	\(\tan{x}\)
8	\(\csc^2{x}\)	\(-\cot{x}\)
9	\(\frac{1}{a^2+x^2}\)	\({1\over a}{\rm Tan}^{-1}{x\over a}\)	\(a>0\)
10	\(\frac{1}{\sqrt{a^2-x^2}}\)	\({\rm Sin}^{-1}{x\over a}\)
11	\(\frac{1}{x^2-a^2}\)	\({1\over2a}\ln\left|\frac{x-a}{x+a}\right|\)
12	\(\frac{1}{\sqrt{x^2+A}}\)	\(\ln\left|x+\sqrt{x^2+A}\right|\)	\(A\ne0\)
13	\(\frac{1}{(x-a)(x-b)}\)	\({1\over a-b}\ln\left|\frac{x-a}{x-b}\right|\)	\(a\ne b\)
14	\(a^x\)	\(\frac{a^x}{\ln{a}}\)	\(a\ne1,\,a>0\)
15	\(\tan{x}\)	\(\ln{|\sec{x}|}\)
16	\(\cot{x}\)	\(-\ln{|\csc{x}|}\)
17	\(\sqrt{a^2-x^2}\)	\({1\over2}\left(x\sqrt{a^2-x^2}+a^2{\rm Sin}^{-1}{x\over a}\right)\)	\(a>0\)
18	\(\sqrt{x^2+A}\)	\({1\over2}\left(x\sqrt{x^2+A}+A\ln{|x+\sqrt{x^2+A}|}\right)\)	\(A\ne0\)
19	\(\frac{f'(x)}{f(x)}\)	\(\ln{|f(x)|}\)
20	\(\{f(x)\}^\alpha f'(x)\)	\(\frac{\{f(x)\}^{\alpha+1}}{\alpha+1}\)	\(\alpha\ne-1\)
21	\(\frac{1}{a^4-x^4}\)	\({1\over2a^3}{\rm Tan}^{-1}{x\over a}+{1\over4a^3}\ln\left|\frac{x+a}{x-a}\right|\)	\(a>0\)