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Fungsi sinus dan kosinus hiperbolik berturut-turut didefinisikan sebagai $\\sinh(x)=\\dfrac{e^x-e^{-x}}{2}$ dan $\\cosh(x)=\\dfrac{e^x+e^{-x}}{2}$. Diberikan $f(x) =\\cosh(\\var{a}x)$ dan $g(x)=\\sinh(\\var{a}x)$.

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Dapat dibuktikan bahwa fungsi $f$ mempunyai invers untuk $x>0$ dan fungsi $g$ mempunyai invers untuk $x\\in\\mathbb{R}$.

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Pilihlah seluruh hubungan yang tepat tentang $f(x)$ dan $g(x)$.

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Diberikan integral $\\displaystyle \\int \\dfrac{\\cosh(x)}{\\sinh^2(x)+\\sinh^4(x)}\\, dx$. Integral tersebut dapat dituliskan dalam bentuk pecahan parsial $\\displaystyle \\int \\cosh(x)\\left(\\dfrac{A}{\\sinh(x)}+\\dfrac{B}{\\sinh^2(x)}+\\dfrac{C\\sinh(x)+D}{\\sinh^2(x)+1}\\right)\\, dx$ dengan $A$, $B$, $C$, dan $D$ berturut-turut adalah $\\ldots$.

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Jika $I(x) = \\displaystyle \\int_{0}^{\\cosh^{-1}(x)} \\dfrac{\\cosh(t)}{\\sinh^2(t)+\\sinh^4(t)}\\, dt$ untuk $x>0$, maka $I\\left(\\sqrt{3}\\right) = 1-\\dfrac{1}{\\sqrt{a}}+\\dfrac{\\pi}{b}$ dengan $a +b =\\ldots$.

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