Question Number 210133 by SANOGO last updated on 31/Jul/24
![calcul ∫_0 ^1 [nt^(n−1) (1−t)−t^n ]dt](https://www.tinkutara.com/question/Q210133.png)
$${calcul} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \left[{nt}^{{n}−\mathrm{1}} \left(\mathrm{1}−{t}\right)−{t}^{{n}} \right]{dt} \\ $$
Answered by mr W last updated on 01/Aug/24
![=∫_0 ^1 (nt^(n−1) −(n+1)t^n )dt =[t^n −t^(n+1) ]_0 ^1 =1^n −1^(n+1) =0](https://www.tinkutara.com/question/Q210140.png)
$$=\int_{\mathrm{0}} ^{\mathrm{1}} \left({nt}^{{n}−\mathrm{1}} −\left({n}+\mathrm{1}\right){t}^{{n}} \right){dt} \\ $$$$=\left[{t}^{{n}} −{t}^{{n}+\mathrm{1}} \right]_{\mathrm{0}} ^{\mathrm{1}} \\ $$$$=\mathrm{1}^{{n}} −\mathrm{1}^{{n}+\mathrm{1}} \\ $$$$=\mathrm{0} \\ $$