## mukushla 3 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 $$f(1)-g(1)$$

1. EulerGroupie

The Rational Zero Theorem suggests that (x-1) 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?

2. helder_edwin

$f(1)-g(1)=\arctan 1$

3. EulerGroupie

Not exactly what I got... but close

4. EulerGroupie

hey... 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!

5. EulerGroupie

arctan 1=pi/4