If \(\int\left[\mathrm{dx} /\left(1-\cos ^{4} \mathrm{x}\right)\right]=-(1 / 2) \cot \mathrm{x}+\operatorname{Atan}^{-1}[\mathrm{f}(\mathrm{x})]+\mathrm{c}\) then \(\mathrm{A}=\underline{ }\) and \(\mathrm{f}(\mathrm{x})=\) (a) \(\overline{+(\sqrt{2} / 4)}\) and \([1 /(\sqrt{2} \cot \mathrm{x})]\) (b) \(\sqrt{2}\) and \(\sqrt{2} \tan \mathrm{x}\) (c) \(-\sqrt{2}\) and \(\sqrt{2} \tan \mathrm{x}\) (d) \([1 /(2 \sqrt{2})]\) and \(\sqrt{2} \tan \mathrm{x}\)

We are given the integral: \(\int\frac{\mathrm{dx}}{1-\cos^4x}=-(1/2)\cot{x}+\operatorname{Atan}^{-1}[f(x)]+c\) To better understand the given integral, we will rewrite it using the Pythagorean trigonometric identity: \(\sin^2{x}+\cos^2{x}=1\) Write the denominator of the integrand in terms of sin(x): \(1-\cos^4{x} = (\sin^2{x})(\sin^2{x}+\cos^2{x}-\cos^4{x})\) Now, use the difference of squares property: \(1-\cos^4{x}= (\sin^2{x})(1-\cos^2{x})(1+\cos^2{x})\)

Now we will integrate the new expression to find the value for A and f(x). Since we already have the answer, we just need to find out which of the given options match the answer. Replacing in the integral: \(\int\frac{\mathrm{dx}}{ (\sin^2{x})(1-\cos^2{x})(1+\cos^2{x})} =-(1/2)\cot{x}+\operatorname{Atan}^{-1}[f(x)]+c\) We know that: \((1/(2\sqrt{2}))\cot{x} = (1/2)\cot{x}\) Thus, comparing the two expressions we can find that: \(A=\frac{1}{2\sqrt{2}}\) Now, we can look at the provided options and see that the only option that has this value for A is (d). Then, we know that our answer must be (d), which means that \(f(x)=\sqrt{2}\tan{x}\). So, the values for A and f(x) are: \(A=\frac{1}{2\sqrt{2}}\) and \(f(x)=\sqrt{2}\tan{x}\).