# Thread: Integrate, Riemann.

1. ## Integrate, Riemann.

Hi, i can't solve this, please if someone can solve and show me the steps (:
$\displaystyle$\displaystyle \int_{1}^{e} \ln (x) dx $$P is the partition. \displaystyle P:[q^{0};q^{1};q^{2};...;q^{n}] ; \displaystyle q=e^{\frac{1}{n}}$$

Thanks for all the answers (:

2. ## Re: Integrate, Riemann.

Originally Posted by SimpleMan
Hi, i can't solve this, please if someone can solve
What do you mean by "solve" here?

Originally Posted by SimpleMan
$\displaystyle$\displaystyle \int_{1}^{e} \ln (x) dx $$P is the partition. \displaystyle P:[q^{0};q^{1};q^{2};...;q^{n}] ; \displaystyle q=e^{\frac{1}{n}}$$
What do you mean by $\displaystyle$[q^{0};q^{1};q^{2};...;q^{n}]$;$\displaystyle q=e^{\frac{1}{n}}?

3. ## Re: Integrate, Riemann.

so let ln(x) be f-1(x) so ex=f(x)
F-1(x)=xf-1(x)-F(f-1(x))
so
$\displaystyle \int_{1}^{e} \ln (x) dx$= x(ln(x)- eln(x) = xln(x)-x evaluated at e and 1

which should be 1

if you haven't seen the equation before look at the geometry between a function and its inverse... they add up to make a rectangle.

I also don't know what the partition thingamabobs are but the answer is 1

4. ## Re: Integrate, Riemann.

Yes, that is easy to integrate but I suspect Simple Man is asking about the Riemann sum that will give that integral. The partition is $\displaystyle e^{0/n}=1, e^{1/n}, e^{2/n}, ..., e^{n/n}= e$. Those values for x make it easy to find the corresponding "f(x)": $\displaystyle ln(1)= 0, ln(e^{1/n})= \frac{1}{n}, ln(e^{2/n})= \frac{2}{n}, ..., ln(e)= 1$. However, it is also makes the interval lengths variable- the length of the first interval is $\displaystyle e^{1/n}- 1$, the second $\displaystyle e^{2/n}- e^{1/n}$, etc.

The Riemann sum will be $\displaystyle \frac{e^{2/n}- e^{1/n}}{n}+ \frac{2(e^{3/n}- e^{2/n})}{n}+ \frac{3(e^{4/n}- e^{3/n})}{n}+ \cdot\cdot\cdot+ \frac{n(e- e^{(n-1)/n}}{n}$ which is, of course, the same as
$\displaystyle \frac{1}{n}(ne- e^{(n-1)/n}- \cdot\cdot\cdot- e^{2/n}+ e^{1/n})$