## anonymous 4 years ago what is "In" in definite integral?

1. anonymous

It's $x \ln x -x$ would you like me to prove it?

2. anonymous

well, xlnx - x evaluated between your two points

3. anonymous

explain a little bit more plz..

4. anonymous

can i send u a unsolved file ?

5. anonymous

Let's prove it then... $\displaystyle\int_{a}^{b}\ln x dx.$ Let u=lnx so that dx=xdu, x=e^u. Then our integral becomes $\displaystyle\int_{\ln a}^{\ln b}ue^udu$ now use integration by parts $[ue^u]_{u= \ln a}^{u=\ln b} - \displaystyle\int_{\ln a}^{\ln b}e^udu$ $= \left[(u-1)e^u\right]_{u=\ln a}^{u=\ln b}$ $=\left[(\ln x -1)x\right]_{x=a}^{x=b}$ $=b(\ln b-1) - a(\ln a -1)$ $=\ln b^b - \ln a^a +(a-b)$ $=\ln \frac{b^b}{a^a} + a -b$

6. anonymous

I can't open it

7. anonymous

why?

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