{"version":9,"randomSeed":"bf506afa8d88cd65da1f380e5adedba7","graph":{"viewport":{"xmin":-26.34144442610696,"ymin":-45.30539211095234,"xmax":12.621190713272398,"ymax":41.22489745246284}},"expressions":{"list":[{"type":"expression","id":"1","color":"#c74440","latex":"\\int_{ }^{ }\\coth udu=\\int_{ }^{ }\\frac{\\cosh u}{\\sinh u}du=\\int_{ }^{ }\\frac{e^{u}+e^{-u}}{e^{u}-e^{-u}}du","labelSize":"medium"},{"type":"expression","id":"4","color":"#6042a6","latex":"w=e^{u},\\ \\ \\int_{ }^{ }\\frac{e^{u}+e^{-u}}{e^{u}-e^{-u}}du=\\int_{ }^{ }\\frac{w+\\frac{1}{w}}{w-\\frac{1}{w}}du=\\int_{ }^{ }\\frac{w^{2}+1}{w^{2}-1}du=\\int_{ }^{ }1+\\frac{2}{w^{2}-1}du=\\int_{ }^{ }1+\\frac{2}{w^{2}-1}du=u+\\int_{ }^{ }\\frac{2}{\\left(w+1\\right)\\left(w-1\\right)}du"},{"type":"expression","id":"6","color":"#c74440","latex":"\\frac{2}{\\left(w+1\\right)\\left(w-1\\right)}=\\frac{1}{w-1}-\\frac{1}{w+1}","labelSize":"medium"},{"type":"expression","id":"7","color":"#2d70b3","latex":"u+\\int_{ }^{ }\\frac{2}{\\left(w+1\\right)\\left(w-1\\right)}du=u+\\int_{ }^{ }\\frac{1}{w-1}du-\\int_{ }^{ }\\frac{1}{w+1}du=u+\\int_{ }^{ }\\frac{1}{e^{u}-1}du-\\int_{ }^{ }\\frac{1}{e^{u}+1}du"},{"type":"expression","id":"10","color":"#000000","latex":"w=e^{u}-1,\\ \\ \\int_{ }^{ }\\frac{1}{e^{u}-1}du=\\int_{ }^{ }\\frac{1}{w}\\cdot\\frac{dw}{\\frac{dw}{du}}=\\int_{ }^{ }\\frac{1}{w}\\cdot\\frac{dw}{w}=\\int_{ }^{ }\\frac{1}{w^{2}}dw=-\\frac{1}{w}+C=-\\frac{1}{e^{u}-1}+C"},{"type":"expression","id":"12","color":"#2d70b3","latex":"w=e^{u}+1,\\ \\ \\int_{ }^{ }\\frac{1}{e^{u}+1}du=\\int_{ }^{ }\\frac{1}{w}\\cdot\\frac{dw}{\\frac{dw}{du}}=\\int_{ }^{ }\\frac{1}{w}\\cdot\\frac{dw}{w}=\\int_{ }^{ }\\frac{1}{w^{2}}dw=-\\frac{1}{w}+C=-\\frac{1}{e^{u}+1}+C"},{"type":"expression","id":"14","color":"#6042a6","latex":"\\int_{ }^{ }\\coth udu=u+\\int_{ }^{ }\\frac{1}{e^{u}-1}du-\\int_{ }^{ }\\frac{1}{e^{u}+1}du=u-\\frac{1}{e^{u}-1}+\\frac{1}{e^{u}+1}+C"}]}}