Question Number 219600 by Nicholas666 last updated on 29/Apr/25
![∫_( 1) ^( 2) [x]^x dx](https://www.tinkutara.com/question/Q219600.png)
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\:\mathrm{1}} ^{\:\mathrm{2}} \left[{x}\right]^{{x}} \:{dx} \\ $$$$ \\ $$
Answered by Nicholas666 last updated on 29/Apr/25
![∫_1 ^( 2) [x]^x dx with;1≤x<2 ,[x]=1 ∫_1 ^( 2) 1^x dx = ∫_1 ^( 2) 1 dx =[x]_1 ^2 =2−1 ⇒1](https://www.tinkutara.com/question/Q219601.png)
$$\int_{\mathrm{1}} ^{\:\mathrm{2}} \left[{x}\right]^{{x}} {dx}\:\:\:\:{with};\mathrm{1}\leqslant{x}<\mathrm{2}\:,\left[{x}\right]=\mathrm{1} \\ $$$$\:\int_{\mathrm{1}} ^{\:\mathrm{2}} \mathrm{1}^{{x}} \:{dx}\:=\:\int_{\mathrm{1}} ^{\:\mathrm{2}} \mathrm{1}\:{dx}\:=\left[{x}\right]_{\mathrm{1}} ^{\mathrm{2}} =\mathrm{2}−\mathrm{1}\:\Rightarrow\mathrm{1} \\ $$