Question

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

Answer 1

`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`