\int_0^\infty \left(\frac{\sinh(ax)}{\sinh(x)}-\frac{a}{e^{2x}}\right)\,\frac{dx}{x}
In this post We will compute the following integral: Then We used that (see here ): and ( here )
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HARD INTEGRAL - PART II
Today We will compute the following integral following the same ideas of the previous post : Recall (see here ) (1) Integrating both sides of (1) from 0 to Computing Recall (see here ) Letting we obtain We are looking for the Imaginary part of the equation above: Computing the quantities: The Glaisher function We know that (see here ): If we integrate from 0 to x we obtain Integrating from 0 to x we obtain
VARIATION OF BINET´S INTEGRAL
In this post we obtain the following integrals We have previously proved the following result (1) Letting we obtain (2) From (2), we can easily obtain the following relations (3) Lemma 1 (4) Proof: From (4) and (3) we obtain In the same fashion we obtain Lemma 2: (5) From Lemma 2 and (3) we obtain: Similarly
INTEGRAL sinh(ax)/sinh(bx)dx
In this post we will prove 4 beautiful integrals involving the hyperbolic sine function, namely (1) Where We used the partial fraction decomposition of the letting proved here A second method Where we used the results proved here and (see appendix below) Now, if we let in (1) we obtain (2) Recall (3) Differentiating both sides of w.r. to setting and Where we used the results proved here and Appendix Recall the reflection formula proved here : Letting we get Taking logs on both sides and differentiating w.t. to x