Inverse Trigonometric Functions - Expression As Definite Integrals

Expression As Definite Integrals

Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:

\begin{align} \arcsin x &{}= \int_0^x \frac {1} {\sqrt{1 - z^2}}\,dz,\qquad |x| \leq 1\\ \arccos x &{}= \int_x^1 \frac {1} {\sqrt{1 - z^2}}\,dz,\qquad |x| \leq 1\\ \arctan x &{}= \int_0^x \frac 1 {z^2 + 1}\,dz,\\ \arccot x &{}= \int_x^\infty \frac {1} {z^2 + 1}\,dz,\\ \arcsec x &{}= \int_1^x \frac 1 {z \sqrt{z^2 - 1}}\,dz, \qquad x \geq 1\\ \arcsec x &{}= \pi + \int_x^{-1} \frac 1 {z \sqrt{z^2 - 1}}\,dz, \qquad x \leq -1\\ \arccsc x &{}= \int_x^\infty \frac {1} {z \sqrt{z^2 - 1}}\,dz, \qquad x \geq 1\\ \arccsc x &{}= \int_{-\infty}^x \frac {1} {z \sqrt{z^2 - 1}}\,dz, \qquad x \leq -1 \end{align}

When x equals 1, the integrals with limited domains are improper integrals, but still well-defined.

Read more about this topic:  Inverse Trigonometric Functions

Famous quotes containing the words expression and/or definite:

    “They are not callow like the young of most birds, but more perfectly developed and precocious even than chickens. The remarkably adult yet innocent expression of their open and serene eyes is very memorable. All intelligence seems reflected in them. They suggest not merely the purity of infancy, but a wisdom clarified by experience. Such an eye was not born when the bird was, but is coeval with the sky it reflects. The woods do not yield another such a gem.”
    —Henry David Thoreau (1817–1862)

    “God is the efficient cause not only of the existence of things, but also of their essence.
    Corr. Individual things are nothing but modifications of the attributes of God, or modes by which the attributes of God are expressed in a fixed and definite manner.”
    —Baruch (Benedict)