On the integration of basic integrals

Published on 2025-01-16.

For most of the functions defined by the Floating type class, their integrals are well-known, and we use what basically amounts to a table lookup to find their integrals. While relying only on these directly would severely limit the variety of functions which can be integrated, these can be used with term by term integration or integration by substitution to significantly expand the range of functions which can be integrated.

This page lists out such functions and their integrals. These are implemented in the following modules:

Integration of powers of the variable

For any expression \(x^n\), except for when \(n=-1\),

\[\int x^n \, dx = \frac{1}{n+1} x^{n+1}\]

\(x^{-1} = \frac{1}{x}\) is the exception to the above pattern.

\[\int \frac{1}{x} \, dx = \log x\]

Integration of exponentials

\[\int e^{x} \, dx = e^{x}\]

Integration of logarithms

\[\int \log x \, dx = -x + \left(\log x\right) x\]

Integration of trigonometric functions

\[\int \sin x \, dx = -\cos x\]\[\int \cos x \, dx = \sin x\]\[\int \tan x \, dx = -\log \left\lvert \cos x \right\rvert\]\[\int \sin^{-1} x \, dx = \left(1 - x^{2}\right)^{\frac{1}{2}} + \left(\sin^{-1} x\right) x\]\[\int \cos^{-1} x \, dx = -\left(1 - x^{2}\right)^{\frac{1}{2}} + \left(\cos^{-1} x\right) x\]\[\int \tan^{-1} x \, dx = -\frac{\log \left(1 + x^{2}\right)}{2} + \left(\tan^{-1} x\right) x\]\[\int \sinh x \, dx = \cosh x\]\[\int \cosh x \, dx = \sinh x\]\[\int \tanh x \, dx = \log \left(\cosh x\right)\]\[\int \sinh^{-1} x \, dx = -\left(1 + x^{2}\right)^{\frac{1}{2}} + \left(\sinh^{-1} x\right) x\]\[\int \cosh^{-1} x \, dx = -\left(-1 + x\right)^{\frac{1}{2}} \left(1 + x\right)^{\frac{1}{2}} + \left(\cosh^{-1} x\right) x\]\[\int \tanh^{-1} x \, dx = \frac{\log \left(1 - x^{2}\right)}{2} + \left(\tanh^{-1} x\right) x\]