Primitive of Reciprocal of Root of x squared plus Constant
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Theorem
- $\ds \int \frac {\d x} {\sqrt {x^2 + k} } = \ln \size {x + \sqrt {x^2 + k} } + C$
Proof
Positive Constant
Let $k > 0$.
Then $k = a^2$ for some $a \in \R$.
Hence from Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$: Logarithm Form:
- $\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \map \ln {x + \sqrt {x^2 + a^2} } + C$
from which the result follows.
$\Box$
Negative Constant
Let $k < 0$.
Then $k = -a^2$ for some $a \in \R$.
Hence from Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$: Logarithm Form:
- $\ds \int \frac {\d x} {\sqrt {x^2 - a^2} } = \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $0 < a < \size x$.
from which the result follows.
$\Box$
Zero Constant
Let $k = 0$.
Then we have:
\(\ds \int \frac {\d x} {\sqrt {x^2 + k} }\) | \(=\) | \(\ds \int \frac {\d x} {\sqrt {x^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\d x} x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size x + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size x + \ln 2 + C'\) | where $C' - C - \ln 2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size {2 x} + C'\) | Sum of Logarithms | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size {x + x} + C'\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size {x + \sqrt {x^2 + 0} } + C'\) |
The result follows.
$\blacksquare$
Sources
- 1945: A. Geary, H.V. Lowry and H.A. Hayden: Advanced Mathematics for Technical Students, Part I ... (previous) ... (next): Chapter $\text {III}$: Integration: Integration