Hyperbolic Function - Standard Integrals

Standard Integrals

For a full list of integrals of hyperbolic functions, see list of integrals of hyperbolic functions.

\begin{align} \int \sinh (ax)\,dx &= a^{-1} \cosh (ax) + C \\ \int \cosh (ax)\,dx &= a^{-1} \sinh (ax) + C \\ \int \tanh (ax)\,dx &= a^{-1} \ln (\cosh (ax)) + C \\ \int \coth (ax)\,dx &= a^{-1} \ln (\sinh (ax)) + C \\ \int \operatorname{sech} (ax)\,dx &= a^{-1} \arctan (\sinh (ax)) + C \\ \int \operatorname{csch} (ax)\,dx &= a^{-1} \ln \left( \tanh \left( \frac{ax}{2} \right) \right) + C
\end{align}
\begin{align} \int {\frac{du}{\sqrt{a^2 + u^2}}} & = \sinh ^{-1}\left( \frac{u}{a} \right) + C \\ \int {\frac{du}{\sqrt{u^2 - a^2}}} &= \cosh ^{-1}\left( \frac{u}{a} \right) + C \\ \int {\frac{du}{a^2 - u^2}} & = a^{-1}\tanh ^{-1}\left( \frac{u}{a} \right) + C; u^2 < a^2 \\ \int {\frac{du}{a^2 - u^2}} & = a^{-1}\coth ^{-1}\left( \frac{u}{a} \right) + C; u^2 > a^2 \\ \int {\frac{du}{u\sqrt{a^2 - u^2}}} & = -a^{-1}\operatorname{sech}^{-1}\left( \frac{u}{a} \right) + C \\ \int {\frac{du}{u\sqrt{a^2 + u^2}}} & = -a^{-1}\operatorname{csch}^{-1}\left| \frac{u}{a} \right| + C
\end{align}

, where C is the constant of integration.

Read more about this topic:  Hyperbolic Function

Famous quotes containing the word standard:

    An indirect quotation we can usually expect to rate only as better or worse, more or less faithful, and we cannot even hope for a strict standard of more and less; what is involved is evaluation, relative to special purposes, of an essentially dramatic act.
    Willard Van Orman Quine (b. 1908)