Primitive of Reciprocal of Square of p plus q by Exponential of a x
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Theorem
- $\ds \int \frac {\d x} {\paren {p + q e^{a x} }^2} = \frac x {p^2} + \frac 1 {a p \paren {p + q e^{a x} } } - \frac 1 {a p^2} \ln \size {p + q e^{a x} } + C$
Proof
\(\ds z\) | \(=\) | \(\ds p + q e^{a x}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d z} {\d x}\) | \(=\) | \(\ds a q e^{a x}\) | Derivative of $e^{a x}$ | ||||||||||
\(\ds \) | \(=\) | \(\ds a \paren {z - p}\) | in terms of $z$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {p + q e^{a x} }\) | \(=\) | \(\ds \int \frac {\d z} {a \paren {z - p} z^2}\) | Integration by Substitution | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {\d z} {z^2 \paren {z - p} }\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \paren {\frac {-1} {-p z} + \frac 1 {\paren {-p}^2} \ln \size {\frac {z - p} z} } + C\) | Primitive of $\dfrac 1 {x^2 \paren {a x + b} }$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a p z} + \frac 1 {a p^2} \ln \size {\frac {z - p} z} + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a p \paren {p + q e^{a x} } } + \frac 1 {a p^2} \ln \size {\frac {q e^{a x} } {p + q e^{a x} } } + C\) | substituting for $z$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a p \paren {p + q e^{a x} } } + \frac 1 {a p^2} \ln \size {q e^{a x} } - \frac 1 {a p^2} \ln \size {p + q e^{a x} } + C\) | Difference of Logarithms | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a p \paren {p + q e^{a x} } } + \frac 1 {a p^2} \ln \size q + \frac 1 {a p^2} \ln \size {e^{a x} } - \frac 1 {a p^2} \ln \size {p + q e^{a x} } + C\) | Sum of Logarithms | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a p \paren {p + q e^{a x} } } + \frac 1 {a p^2} \ln \size q + \frac 1 {a p^2} \map \ln {e^{a x} } - \frac 1 {a p^2} \ln \size {p + q e^{a x} } + C\) | $e^{a x}$ always positive | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a p \paren {p + q e^{a x} } } + \frac 1 {a p^2} \map \ln {e^{a x} } - \frac 1 {a p^2} \ln \size {p + q e^{a x} } + C\) | $\dfrac 1 {a p^2} \ln \size q$ subsumed into constant | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a p \paren {p + q e^{a x} } } + \frac 1 {a p^2} a x - \frac 1 {a p^2} \ln \size {p + q e^{a x} } + C\) | Exponential of Natural Logarithm | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac x {p^2} + \frac 1 {a p \paren {p + q e^{a x} } } - \frac 1 {a p^2} \ln \size {p + q e^{a x} } + C\) | simplification |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $e^{a x}$: $14.516$