INTEGRALS CONTAINING $$\frac{1}{\sinh ax}$$

1.

$$\int\displaystyle \frac{dx}{\sinh ax}=\displaystyle \frac{1}{a}\ln\tanh\displaystyle \frac{ax}{2}$$

2.

$$\int\displaystyle \frac{dx}{\sinh^2 ax}=-\displaystyle \frac{\coth ax}{a}$$

3.

$$\int\displaystyle \frac{dx}{\sinh^3 ax}=-\displaystyle \frac{\coth ax}{2a\sinh ax}-\displaystyle \frac{1}{2a}\ln\tanh\displaystyle \frac{ax}{2}$$

4.

$$\int\displaystyle \frac{\coth ax}{\sinh^n ax}dx=-\displaystyle \frac{1}{na\sinh^n ax}$$

5.

$$\int\sinh ax dx=\displaystyle \frac{1}{a}\cosh ax$$

6.

$$\int \displaystyle \frac{x dx}{\sinh ax}=\displaystyle \frac{1}{a^... ...tyle \frac{2(-1)^n(2^{2n-1}-1)B_n(ax)^{2n+1}}{(2n+1)!}+\cdot\cdot\cdot \right\}$$

where the constants B_n are the Bernoulli's numbers.

7.

$$\int\displaystyle \frac{x dx}{\sinh^2 ax}=-\displaystyle \frac{x\coth ax}{a}+\displaystyle \frac{1}{a^2}\ln\sinh ax$$

8.

$$\int\displaystyle \frac{dx}{x\sinh ax}=-\displaystyle \frac{1}{ax}... ...aystyle \frac{(-1)^n 2(2^{2n-1}-1)B_n(ax)^{2n-1}}{(2n-1)(2n)!}+\cdot\cdot\cdot$$

where the constants B_n are the Bernoulli's numbers.

9.

$$\int\displaystyle \frac{dx}{q+p/\sinh ax}=\displaystyle \frac{x}{q}-\displaystyle \frac{p}{q}\int\displaystyle \frac{dx}{p+q/\sinh ax}$$

10.

$$\int\displaystyle \frac{dx}{\sinh^n ax}=\displaystyle \frac{-\coth... ...2} ax}-\displaystyle \frac{n-2}{n-1}\int\displaystyle \frac{dx}{\sinh^{n-2} ax}$$

[Tables]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Copyright © 1999-2024 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour