# Engineering Tables/Table of Integrals

Jump to: navigation, search

This is a small summary of the identities found Here.

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