1. ## how to integrate?

If a function f is defined by f(x) = S from 0 to x (1 / (1 + t^4)) dt, is f(1) = to 1/2?

I take it I have to integrate that thing, how do I do it?

2. Originally Posted by LeoBloom.
If a function f is defined by f(x) = S from 0 to x (1 / (1 + t^4)) dt, is f(1) = to 1/2?

I take it I have to integrate that thing, how do I do it?
we can say no without actually evaluating the integral.

$\displaystyle f(1)=\int_{0}^{1}\frac{1}{1+t^4}dt > \frac{1}{2}$

By camparison the minimum of $\displaystyle \frac{1}{1+t^4}$ on [0,1] is 1/2 and the max is 1 so

$\displaystyle \int_{0}^{1}\frac{1}{2}dt < \int_{0}^{1}\frac{1}{1+t^4}dt < \int_{0}^{1}1dt$

equallity could only happen if the integrand was constant on [0,1]

I hope this helps.

Good luck.

3. Nope, sorry I didn't follow. I dont see how you get a min of 1/2 and a max of 1.

4. Originally Posted by LeoBloom.
Nope, sorry I didn't follow. I dont see how you get a min of 1/2 and a max of 1.
let

$\displaystyle h(t)=\frac{1}{1+t^4}$

$\displaystyle h(0)=\frac{1}{1+0^4}=1$

$\displaystyle h(1)=\frac{1}{1+1^4}=\frac{1}{2}$

If your really want to be rigorous you could verify this with the first and second derivative test, but the function is postitive and decreasing for all t > 0 so its max is at the left end point and its min is at the right end point.

I hope this clears it up.