Primitive of Reciprocal of p plus q by Hyperbolic Sine of a x
Jump to navigation
Jump to search
Theorem
- $\ds \int \frac {\d x} {p + q \sinh a x} = \frac 1 {a \sqrt{p^2 + q^2} } \ln \size {\frac {q e^{a x} + p - \sqrt {p^2 + q^2} } {q e^{a x} + p + \sqrt {p^2 + q^2} } } + C$
Proof
Let:
\(\ds u\) | \(=\) | \(\ds e^{a x}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {\d u} {\d x}\) | \(=\) | \(\ds e^{a x} = u\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \d x\) | \(=\) | \(\ds \dfrac {\d u} u\) |
Hence:
\(\ds \int \frac {\d x} {p + q \sinh a x}\) | \(=\) | \(\ds \int \frac {\d x} {p + q \paren {\dfrac {e^{a x} - e^{-a x} } 2} }\) | Definition of Real Hyperbolic Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {2 \rd x} {2 p + q \paren {e^{a x} - e^{-a x} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {2 \rd u} {u \paren {2 p + q u - \frac q u} }\) | Integration by Substitution: $u = e^{a x}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {2 \rd u} {q u^2 + 2 p u - q}\) | simplifying |
The discriminant of $q u^2 + 2 p u - q$ is given by:
\(\ds \map {\operatorname {Disc} } {q u^2 + 2 p u - q}\) | \(=\) | \(\ds \paren {2 p}^2 + 4 q^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 \paren {p^2 + q^2}\) |
Hence the sign of $\map {\operatorname {Disc} } {q u^2 + 2 p u - q}$ is always positive.
So:
\(\ds \int \frac {2 \rd u} {q u^2 + 2 p u - q}\) | \(=\) | \(\ds \dfrac 2 {\sqrt {4 p^2 + 4 q^2} } \ln \size {\dfrac {2 q u + 2 p - \sqrt {4 p^2 + 4 q^2} } {2 q u + 2 p + \sqrt {4 p^2 + 4 q^2} } } + C\) | Primitive of $\dfrac 1 {a x^2 + b x + c}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {\sqrt {p^2 + q^2} } \ln \size {\dfrac {q u + p - \sqrt {p^2 + q^2} } {q u + p + \sqrt {p^2 + q^2} } } + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {\sqrt {p^2 + q^2} } \ln \size {\dfrac {q e^{a x} + p - \sqrt {p^2 + q^2} } {q e^{a x} + p + \sqrt {p^2 + q^2} } } + C\) | substituting $u = e^{a x}$ |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x$: $14.553$