Inverse Trigonometric Functions - Indefinite Integrals of Inverse Trigonometric Functions

Indefinite Integrals of Inverse Trigonometric Functions

For real and complex values of x:

\begin{align} \int \arcsin x\,dx &{}= x\,\arcsin x + \sqrt{1-x^2} + C\\ \int \arccos x\,dx &{}= x\,\arccos x - \sqrt{1-x^2} + C\\ \int \arctan x\,dx &{}= x\,\arctan x - \frac{1}{2}\ln\left(1+x^2\right) + C\\ \int \arccot x\,dx &{}= x\,\arccot x + \frac{1}{2}\ln\left(1+x^2\right) + C\\ \int \arcsec x\,dx &{}= x\,\arcsec x - \ln\left + C\\ \int \arccsc x\,dx &{}= x\,\arccsc x + \ln\left + C \end{align}

For real x ≥ 1:

\begin{align} \int \arcsec x\,dx &{}= x\,\arcsec x - \ln\left(x+\sqrt{x^2-1}\right) + C\\ \int \arccsc x\,dx &{}= x\,\arccsc x + \ln\left(x+\sqrt{x^2-1}\right) + C \end{align}

All of these can be derived using integration by parts and the simple derivative forms shown above.

Read more about this topic:  Inverse Trigonometric Functions

Famous quotes containing the words indefinite, inverse and/or functions:

    “... indefinite visions of ambition are weak against the ease of doing what is habitual or beguilingly agreeable.”
    —George Eliot [Mary Ann (or Marian)

    “Yet time and space are but inverse measures of the force of the soul. The spirit sports with time.”
    —Ralph Waldo Emerson (1803–1882)

    “The English masses are lovable: they are kind, decent, tolerant, practical and not stupid. The tragedy is that there are too many of them, and that they are aimless, having outgrown the servile functions for which they were encouraged to multiply. One day these huge crowds will have to seize power because there will be nothing else for them to do, and yet they neither demand power nor are ready to make use of it; they will learn only to be bored in a new way.”
    —Cyril Connolly (1903–1974)