Section B

Appendix B: useful equivalences

Some antiderivatives can be presented in different forms that differ only by a constant.

arsenh (\frac{x}{a}) = ln x + x^{2} + a^{2} - ln a, a > 0

Therefore,

\int \frac{d x}{x ^{2} + a ^{2}} = arsenh (\frac{x}{a}) + C

is equivalent to the logarithmic form.

Likewise,

\int tan x d x = - ln ∣ cos x ∣ + C = ln ∣ sec x ∣ + C .