# Thread: More Anti-Derivatives

1. ## More Anti-Derivatives

I need to find the most common anti-derivative of f '' (x) = x^-2 , where x > 0, f(1) = 0 and f(2) = 0

As usual, this is how far I've gotten:

f'(x) = - x^-1 + C
f(x) = -Inx + Cx + D

I think that's right. Now what am I supposed to do?

I tried finding f(1) and f(2) of the function f(x) but it's not coming out as it should.

The answer is supposed to be f(x) = -Inx + (In2)x - In2

2. Originally Posted by zachb
I need to find the most common anti-derivative of f '' (x) = x^-2 , where x > 0, f(1) = 0 and f(2) = 0

As usual, this is how far I've gotten:

f'(x) = - x^-1 + C
f(x) = -Inx + Cx + D

I think that's right. Now what am I supposed to do?

I tried finding f(1) and f(2) of the function f(x) but it's not coming out as it should.

The answer is supposed to be f(x) = -Inx + (In2)x - In2
You came up with:
f(x) = -Inx + Cx + D
=> f(1) = -ln(1) + C + D = 0
=> C + D = 0 ...................(1) since, ln(1) = 0

=> f(2) = -ln(2) + 2C + D = 0
=> 2C + D = ln(2).............(2)

So now to find C and D, we must solve the system:
C + D = 0 ...................(1)
2C + D = ln(2) .............(2)

=> C = ln(2) ...........(2) - (1)

but C + D = 0
=> ln(2) + D = 0
=> D = -ln(2)

so f(x) = -lnx + ln(2)x - ln(2)

3. Originally Posted by zachb
I need to find the most common anti-derivative of f '' (x) = x^-2 , where x > 0, f(1) = 0 and f(2) = 0

As usual, this is how far I've gotten:

f'(x) = - x^-1 + C
f(x) = -Inx + Cx + D

I think that's right. Now what am I supposed to do?

I tried finding f(1) and f(2) of the function f(x) but it's not coming out as it should.

The answer is supposed to be f(x) = -Inx + (In2)x - In2
Hello,

all your calculations are correct!

You only have to plug in the values you know into the equation of f:

f(1) = 0 = -ln(1) + C*1 + D ===> 0 = C + D (note that ln(1) = 0)
f(2) = 0 = -ln(2) + C*2 * D ===> 0 = -ln(2) + 2C + D

Subtract equ1 from equ2 and you'll get:

0 = -ln(2) + C ===> C = ln(2). Plug in this value into the first equation D = -ln(2).

The equation of f becomes:

f(x) = -ln(x) +(ln(2))x - ln(2)