determine a, b, c if F'=f f=2x/(x+1)(x^2+1) F=a ln (1+x) + b ln ( 1+x^2) + c arctanx

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f= 2x/(x+1)(x^2 +1)

F = aln (1+x) + b ln(1+x^2) +c*arctanx

F' = a/(1+x) +2bx/(1+x^2) + c/(1+x^2)

    = [a(1+x^2) + 2bx(1+x) + c(1+x)]/(1+x^2)(1+x)

    = ( a + ax^2 + 2bx + 2bx^2 + c + x )/(1+x^2)(1+x)

     = [(a+2b)x^2 + (2b+1)x + (a+c)]/(1+x^2)(1+x)

But F' = f

...

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f= 2x/(x+1)(x^2 +1)

F = aln (1+x) + b ln(1+x^2) +c*arctanx

F' = a/(1+x) +2bx/(1+x^2) + c/(1+x^2)

    = [a(1+x^2) + 2bx(1+x) + c(1+x)]/(1+x^2)(1+x)

    = ( a + ax^2 + 2bx + 2bx^2 + c + x )/(1+x^2)(1+x)

     = [(a+2b)x^2 + (2b+1)x + (a+c)]/(1+x^2)(1+x)

But F' = f

==> (a+2b)x^2 + (2b+1)x + (a+c) = 2x

==> a+2b = 0......(1)

==> 2b+1 = 2

==> b = 1/2

==> a = -2b = -2*1/2 = -1

==> a = -1

==> a+c = 0.........(3)

==> c = 1

 

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