What is f(x) = int (x+1)/((x+5)(x-4) ) dx if f(2)=1 ?

1 Answer
May 7, 2018

f(x) = 1+ (4 ln abs(x+5)+5 ln abs(x-4))/9 - (4 ln 7+5 ln 2)/9

Explanation:

Decompose the rational function in partial fractions:

(x+1)/((x+5)(x-4)) = A/(x+5)+B/(x-4)

(x+1)/((x+5)(x-4)) = (A(x-4)+B(x+5))/((x+5)(x-4))

x+1 = (A+B)x -4A+5B

{(A+B=1),(-4A+5B =1):}

{(A=4/9),(B=5/9):}

Then:

int (x+1)/((x+5)(x-4))dx = 4/9 intdx/(x+5)+5/9 int dx/(x-4)

int (x+1)/((x+5)(x-4))dx = 4/9 ln abs(x+5)+5/9 ln abs(x-4)+C

Now, if f(2) = 1 then:

f(x) = 1+int_2^x (t+1)/((t+5)(t-4))dt

f(x) = 1+ (4 ln abs(x+5)+5 ln abs(x-4))/9 - (4 ln 7+5 ln 2)/9