What is f(x) = int (3-x)/(x-4) if f(5)=1 ?

1 Answer
Jan 23, 2017

f(x) = 6 - x - ln|x- 4|

Explanation:

f(x) = int(-(x - 3))/(x- 4)dx

f(x) = -int(x - 3)/(x - 4)dx

Decompose into partial fractions.

A/1 + B/(x -4) = (x - 3)/(1(x - 4))

A(x- 4) + B(1) = x - 3

Ax - 4A + B = x - 3

Write a system of equations:

{(A = 1), (B - 4A = -3):}

Solve:

A = 1 and B = 1.

f(x) =- int 1 + 1/(x - 4)dx

This can be integrated using int(1/x) = ln|x| + C and int(x^n)dx = x^(n +1)/(n + 1) + C.

f(x) = -x - ln|x- 4| + C

Solve for C now.

1 = -5 - ln|5 - 4| + C

1 = -5 + 0 + C

C = 6

The function is therefore f(x) = 6 - x - ln|x- 4|.

Hopefully this helps!