Integrate (x+2)/((x+5)(x-7)) using partial fractions between the limits 5 and -2, giving your answer to 3sf

First, we're going to split this 1 fraction into 2 fractions. Let (x+2)/((x+5)(x-7)) = A/(x+5) + B/(x-7) then multiplying by (x+5)(x-7) gives A(x - 7) + B(x + 5) = x + 2 multiplying this out we achieve Ax - 7A + Bx + 5B = x + 2. By looking at the coefficients of x we can see that Ax + Bx = x, divide by the x as it's common on both sides gives A + B = 1. We can look at our constant coefficients 5B - 7A = 2 and then we have simultaneous equations (multiplying the 1st equation by 7 and adding it to the second gives 12B = 9 therefore B=3/4 and A=1/4) this puts our integration equation into a much simpler form.(x+2)/((x+5)(x-7)) = 1/4(x+5) + 3/4(x-7) when we integrate this we get 1/4(ln|x+5|)+3/4(ln|x-7|), now we can substitute 5 and -2 into the equation 1/4(ln|5+5|)+3/4(ln|5-7|) - ( 1/4(ln|-2+5|)+3/4(ln|-2-7|) ) gives 1/4(ln10)+3/4(ln2) - 1/4(ln3) - 3/4(ln9) = -0.827 to 3sf