1
vote
1answer
27 views

Disappearing negative signs when evaluating a sinh^-1 integral

$$\int_{-2}^{6} \frac{1}{\sqrt{1+(-x)^2}} \, dx$$ When performing this integral on paper, I get $$\sinh^{-1}(6) - \sinh^{-1}(-2) $$ But when I type it on wolframalpha, I get the unintuitive answer ...
2
votes
1answer
55 views

definite integral involing hyperbolic and trigonometric functions

Trying to prove the following: $$ \int_0^\infty xe^{-c x^2}\sinh(a x)\cos(bx)\,dx = ...
2
votes
0answers
64 views

Closed form for $\int^1_0 \mathrm{arccsch}\left( \frac{x^2-x-1}{x^2+x+1}\right)\;\mathrm dx$

Find a closed form for $$\int^1_0 \mathrm{arccsch}\left( \frac{x^2-x-1}{x^2+x+1}\right)\;\mathrm dx$$ What I have tried Expanding the $\mathrm{arccsch}$ into its logarithmic form, however I ...
13
votes
3answers
263 views

Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$

Let $$f(a)=\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx,$$ where $\operatorname{sech}(z)=\frac2{e^z+e^{-z}}$ is the hyperbolic secant. Here are values of $f(a)$ at some ...
2
votes
1answer
136 views

Evaluation of an integral involving hyperbolic sine and exponential

I am wondering if the following integral can be reduced to either a closed form involving elementary functions, or well-known special functions (such as $\operatorname{erf}$, Bessel functions, etc.): ...
5
votes
1answer
135 views

Integral with hyperbolic cosine squared

Does anyone can give me a hint how to integrate the following: $$\int_0^\infty{\frac{x^2 {\rm d}x}{\mathrm{cosh}^2(x)}}.$$ The answer is $\frac{\pi^2}{12}$ (taken from the book). I've started with ...