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:

    “Freedom is but the possibility of a various and indefinite activity; while government, or the exercise of dominion, is a single, yet real activity. The longing for freedom, therefore, is at first only too frequently suggested by the deep-felt consciousness of its absence.”
    —Karl Wilhelm Von Humboldt (1767–1835)

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

    “Mark the babe
    Not long accustomed to this breathing world;
    One that hath barely learned to shape a smile,
    Though yet irrational of soul, to grasp
    With tiny finger—to let fall a tear;
    And, as the heavy cloud of sleep dissolves,
    To stretch his limbs, bemocking, as might seem,
    The outward functions of intelligent man.”
    —William Wordsworth (1770–1850)