Cho hàm số \(f\left( x \right)={{\left| x \right|}^{2021}}\). Giá trị của \(I=\int\limits_{0}^{\frac{\pi }{2}}{f\left( 2\cos x-1 \right)\sin x\text{d}x}\) bằng:

* Hướng dẫn giải

+ Đặt \(t=2\cos x-1\Rightarrow \text{dt}=-2\sin x\text{d}x\Rightarrow \sin x\text{d}x=-\frac{\text{dt}}{2}\).

+ Khi x=0 thì t=1

Khi \(x=\frac{\pi }{2}\) thì t=-1

 + Do đó: \(I=\frac{1}{2}\int\limits_{-1}^{1}{f(t)\text{dt}=}\frac{1}{2}\int\limits_{-1}^{1}{f(x)\text{d}x=\frac{1}{2}\int\limits_{-1}^{1}{{{\left| x \right|}^{2021}}\text{d}x}}\).

Vì \(f(x)={{\left| x \right|}^{2021}}\) là hàm số chẵn nên \(\int\limits_{-1}^{1}{{{\left| x \right|}^{2021}}dx=}2\int\limits_{0}^{1}{{{\left| x \right|}^{2021}}dx}\).

Suy ra \(I=\int\limits_{0}^{1}{{{\left| x \right|}^{2021}}dx=\int\limits_{0}^{1}{{{x}^{2021}}dx=}}\left. \frac{{{x}^{2022}}}{2022} \right|_{0}^{1}=\frac{1}{2022}\).