Primitive of Reciprocal of Root of x squared plus a squared/Inverse Hyperbolic Sine Form
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Theorem
- $\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \arsinh {\frac x a} + C$
Proof
Let:
\(\ds u\) | \(=\) | \(\ds \arsinh {\frac x a}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds a \sinh u\) | Definition of Real Area Hyperbolic Sine | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d x} {\d u}\) | \(=\) | \(\ds a \cosh u\) | Derivative of Hyperbolic Sine | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {\sqrt {x^2 + a^2} }\) | \(=\) | \(\ds \int \frac {a \cosh u} {\sqrt {a^2 \sinh^2 u + a^2} } \rd u\) | Integration by Substitution | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac a a \int \frac {\cosh u} {\sqrt {\sinh^2 u + 1} } \rd u\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\cosh u} {\cosh u} \rd u\) | Difference of Squares of Hyperbolic Cosine and Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \int 1 \rd u\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds u + C\) | Integral of Constant | |||||||||||
\(\ds \) | \(=\) | \(\ds \arsinh {\frac x a} + C\) | Definition of $u$ |
$\blacksquare$
Also see
- Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$: Inverse Hyperbolic Cosine Form
- Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$
Sources
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- 1945: A. Geary, H.V. Lowry and H.A. Hayden: Advanced Mathematics for Technical Students, Part I ... (previous) ... (next): Chapter $\text {III}$: Integration: Integration
- 1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Standard Forms: $\text {(x)}$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.43$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.182$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Front endpapers: A Brief Table of Integrals: $20$.
- 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $7$: Integrals
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $8$: Integrals