Primitive of Reciprocal of q plus p by Hyperbolic Cosecant of a x

Theorem

$\ds \int \frac {\d x} {q + p \csch a x} = \frac x q - \frac p q \int \frac {\d x} {p + q \sinh a x} + C$

Proof

\(\ds \int \frac {\d x} {q + p \csch a x}\)

\(=\)

\(\ds \frac 1 q \int \frac {q \rd x} {q + p \csch a x}\)

multiplying top and bottom by $q$

\(\ds \)

\(=\)

\(\ds \frac 1 q \int \frac {\paren {q + p \csch a x - p \csch a x} \rd x} {q + p \csch a x}\)

\(\ds \)

\(=\)

\(\ds \frac 1 q \int \frac {\paren {q + p \csch a x} \rd x} {q + p \csch a x} - \frac p q \int \frac {\csch a x \rd x} {q + p \csch a x}\)

Linear Combination of Primitives

\(\ds \)

\(=\)

\(\ds \frac 1 q \int \d x - \frac p q \int \frac {\csch a x \rd x} {q + p \csch a x}\)

simplifying

\(\ds \)

\(=\)

\(\ds \frac x q - \frac p q \int \frac {\csch a x \rd x} {q + p \csch a x} + C\)

Primitive of Constant

\(\ds \)

\(=\)

\(\ds \frac x q - \frac p q \int \frac {\d x} {\frac q {\csch a x} + p} + C\)

multiplying top and bottom by $\dfrac 1 {\csch a x}$

\(\ds \)

\(=\)

\(\ds \frac x q - \frac p q \int \frac {\d x} {p + q \sinh a x} + C\)

Definition 2 of Hyperbolic Cosecant

$\blacksquare$

Also see

• Primitive of $\dfrac 1 {p + q \sinh a x}$

• Primitive of $\dfrac 1 {p + q \cosh a x}$

• Primitive of $\dfrac 1 {p + q \tanh a x}$

• Primitive of $\dfrac 1 {p + q \coth a x}$

• Primitive of $\dfrac 1 {q + p \sech a x}$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\csch a x$: $14.644$