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:

    Every word we speak is million-faced or convertible to an indefinite number of applications. If it were not so we could read no book. Your remark would only fit your case, not mine.
    Ralph Waldo Emerson (1803–1882)

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

    When Western people train the mind, the focus is generally on the left hemisphere of the cortex, which is the portion of the brain that is concerned with words and numbers. We enhance the logical, bounded, linear functions of the mind. In the East, exercises of this sort are for the purpose of getting in tune with the unconscious—to get rid of boundaries, not to create them.
    Edward T. Hall (b. 1914)