The discussion centers on solving the integral of the function involving hyperbolic sine and cosine, specifically the expression $$\int\dfrac{6\sinh(x)\cosh^2(x)+\sinh(x)}{\cosh^3(x)+\cosh(x)}\,dx$$. The integral is simplified using the identity for hyperbolic tangent, leading to the expression $$\int\tanh(x)\left(\dfrac{5\cosh^2(x)}{\cosh^2(x)+1}+1\right)\,dx$$. Participants are encouraged to continue solving the integral by breaking it down into manageable parts, including $$5\int\dfrac{\cosh(x)\sinh(x)}{\cosh^2(x)+1}\,dx$$ and $$\int\tanh(x)\,dx$$.
PREREQUISITESStudents and professionals in mathematics, particularly those studying calculus and integral equations, as well as educators looking for examples of hyperbolic function integration.
\text{We have: }\;\int \left(\frac{3\sinh x\cosh^2\!x + \sinh x}{\cosh^3\!x + \cosh x} + \frac{3\sinh x\cosh^2\!x}{\cosh^3\!x + \cosh x}\right)\,dxmahmoud shaaban said:
