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:

    “The American adolescent, then, is faced, as are the adolescents of all countries who have entered or are entering the machine age, with the question: freedom from what and at what price? The American feels so rich in his opportunities for free expression that he often no longer knows what it is he is free from. Neither does he know where he is not free; he does not recognize his native autocrats when he sees them.”
    —Erik H. Erikson (1904–1994)

    “There are many definite methods, honest and dishonest, which make people rich; the only “instinct” I know of which does it is that instinct which theological Christianity crudely describes as “the sin of avarice.””
    —Gilbert Keith Chesterton (1874–1936)