AcademyIntegration
Academy
Standard Integrals
Level 1 - Math I (Physics) topic page in Integration.
Standard Integrals Reference Table
These are the fundamental integrals that form the building blocks for solving more complex integrals.
Basic Power Rule
Power
\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)\]
Exponential Functions
Exponential
\[\int e^x \, dx = e^x + C\]
Exponential Power
\[\int a^x \, dx = \frac{a^x}{\ln a} + C \quad (a > 0, a \neq 1)\]
Logarithmic Functions
Log
\[\int \frac{1}{x} \, dx = \ln|x| + C\]
Log Base
\[\int \log_a x \, dx = \frac{x \ln x - x}{\ln a} + C\]
Trigonometric Functions
Sin
\[\int \sin x \, dx = -\cos x + C\]
Cos
\[\int \cos x \, dx = \sin x + C\]
Sec²
\[\int \sec^2 x \, dx = \tan x + C\]
Csc²
\[\int \csc^2 x \, dx = -\cot x + C\]
Sec Tan
\[\int \sec x \tan x \, dx = \sec x + C\]
Csc Cot
\[\int \csc x \cot x \, dx = -\csc x + C\]
Inverse Trigonometric Functions
Arcsin
\[\int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin x + C\]
Arctan
\[\int \frac{1}{1+x^2} \, dx = \arctan x + C\]
Arcsec
\[\int \frac{1}{|x|\sqrt{x^2-1}} \, dx = \operatorname{arcsec}|x| + C\]
Trigonometric Substitutions
| Expression | Substitution |
|---|---|
| \(\sqrt{a^2 - x^2}\) | \(x = a \sin\theta\) |
| \(\sqrt{a^2 + x^2}\) | \(x = a \tan\theta\) |
| \(\sqrt{x^2 - a^2}\) | \(x = a \sec\theta\) |
These standard forms and substitution patterns enable integration of many rational and radical expressions.