Definite Integral of Reciprocal of Root of a Squared minus x Squared
Jump to navigation
Jump to search
It has been suggested that this page or section be merged into Arcsine as Integral. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mergeto}} from the code. |
Theorem
- $\ds \int_0^x \frac {\d t} {\sqrt{1 - t^2} } = \arcsin x$
Proof
\(\ds \int_0^x \frac {\d t} {\sqrt{1 - t^2} }\) | \(=\) | \(\ds \intlimits {\arcsin \frac t 1} 0 x\) | Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$, Definition of Definite Integral | |||||||||||
\(\ds \) | \(=\) | \(\ds \arcsin x - \arcsin 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \arcsin x\) |
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.33$: Weierstrass ($\text {1815}$ – $\text {1897}$)