# G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)

1. Apr 13, 2012

### IntegrateMe

G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)....

Find a formula for G(x) in terms of F(x).

I would have just written $$\int_0^x{F(x)dx}$$ but I know the value G(1) = 1 has to be included somewhere. Can someone help me figure this one out?

2. Apr 13, 2012

### Dick

Re: G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)....

How about if you just add a constant?

3. Apr 13, 2012

### IntegrateMe

Re: G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)....

$$\int_0^x{F(x)dx} + C$$??

4. Apr 13, 2012

### Dick

Re: G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)....

Sure. If that is your G(x) then what value of C will make G(1)=1? Put x=1 and solve for C.

5. Apr 13, 2012

### IntegrateMe

Re: G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)....

Well, wouldn't I need to know what the value of $$\int_0^1{F(x)dx}$$ is to determine that?

6. Apr 13, 2012

### Dick

Re: G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)....

No. You can't compute it. $$\int_0^1{F(x)dx}$$ should appear as part of your answer.

7. Apr 13, 2012

### IntegrateMe

Re: G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)....

$$G(x) = 1 + \int_1^xF(s)ds$$ ?

8. Apr 13, 2012

### Dick

Re: G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)....

Good job. Why the '?'

9. Apr 13, 2012

### IntegrateMe

Re: G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)....

I wasn't sure of the limits of integration. They were kind of unjustified.

10. Apr 13, 2012

### Dick

Re: G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)....

If you'd followed the line I was suggesting you should have found $$C=1-\int_0^1{F(s)ds}$$

Can you justify the bounds using that?

11. Apr 13, 2012

### IntegrateMe

Re: G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)....

The F(x) was making it confusing.

12. Apr 13, 2012

### Dick

Re: G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)....

Maybe. If one of the limits of integration is x you shouldn't write the integral dx. Since the integration variable is a dummy you should use something besides x, like s. Makes life less confusing.

13. Apr 13, 2012

### IntegrateMe

Re: G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)....

Can you explain why C would allow you to justify the bounds.

14. Apr 13, 2012

### Dick

Re: G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)....

The integral from 0 to x of F minus the integral from 0 to 1 of F equals the integral from 1 to x of F.