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

1 Answer
Nov 15, 2016

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!