Use integration by partial fractions.
A/(x + 1) + B/(x- 4) = (x - 2)/((x + 1)(x - 4))
A(x - 4) + B(x + 1) = x- 2
Ax - 4A + Bx + B = x - 2
(A + B)x + (B - 4A) = x - 2
We can hence write the following system of equations.
{(A + B = 1), (B - 4A = -2):}
Solve:
B = 1-A
1 - A - 4A = -2
-5A = -3
A = 3/5
A + B = 1
3/5 + B = 1
B = 2/5
Hence, the partial fraction decomposition is 3/(5(x + 1)) + 2/(5(x - 4)). We integrate using the rule int(1/x)dx = ln|x| + C.
=>3/5ln|x + 1| + 2/5ln|x - 4| + C
The function is y= 3/5ln|x + 1| + 2/5ln|x - 4| + C. We know an input/output of the function, so in this case we will solve for C to find the specific function.
We have that when x =2, y = 5.
5 = 3/5ln|2 + 1| + 2/5ln|2 - 4| + C
5 = 3/5ln3 + 2/5ln2 + C
C = 5 - 3/5ln3 - 2/5ln2
C~=4.06
:.The final function is y = 3/5ln|x + 1| + 2/5ln|x - 4| + 4.06, nearly.
Hopefully this helps!