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 2 years ago
if\[f'(x)=\frac{1}{1+x+x^2+x^3}\],\[g'(x)=\frac{x}{1+x+x^2+x^3}\]and \(f(0)=g(0)\)
Find value of \(f(1)g(1)\)
 2 years ago
if\[f'(x)=\frac{1}{1+x+x^2+x^3}\],\[g'(x)=\frac{x}{1+x+x^2+x^3}\]and \(f(0)=g(0)\) Find value of \(f(1)g(1)\)

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EulerGroupie
 2 years ago
Best ResponseYou've already chosen the best response.1The Rational Zero Theorem suggests that (x1) or (x+1) may be factors of the denominators. Using polynomial long division (or synthetic division) indicates that (x+1) is a factor. Use the resulting factored denominator with partial fraction decomposition to break both functions into something that can be integrated. Integrate both functions with separately marked added constants (I used subscript f and g). Set f(x)=g(x) substituting x=0 in for both functions to find what the constants should be. You won't find the exact constants, but you will see how they relate to each other. Now substitute x=1 into f(x)g(x). Would you like the thrill of discovery, or would you like to see my details?

helder_edwin
 2 years ago
Best ResponseYou've already chosen the best response.1\[ f(1)g(1)=\arctan 1\]

EulerGroupie
 2 years ago
Best ResponseYou've already chosen the best response.1Not exactly what I got... but close

EulerGroupie
 2 years ago
Best ResponseYou've already chosen the best response.1hey... I think you are right...arctan x with x=1... yup. I incorrectly put x^2+1 into my argument, but it should be just x...nice!
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