tích phân đặc biệt 3

Question

$\int\limits_{-1/2}^{1/2}cosx.ln\frac{1-x}{1+x}dx$

$\int\limits_{-1}^{1}\frac{x^{4} + x^{2}}{x^{2}+1}dx$

score 6 · Answer 1

Đặt: $I=\int\limits_{-1/2}^{1/2}\cos x\ln\frac{1-x}{1+x}dx$
Đặt: $x=-t\Rightarrow dx=-dt$
Ta có:
$I=-\int\limits_{1/2}^{-1/2}\cos(-t)\ln\frac{1+t}{1-t}dt$
$=\int\limits_{-1/2}^{1/2}\cos t\ln\frac{1+t}{1-t}dt$
$=-\int\limits_{-1/2}^{1/2}\cos t\ln\frac{1-t}{1+t}dt=-I$
Suy ra: $2I=0\Leftrightarrow I=0$

score 6 · Answer 2

Bình chọn tăng 6 Bình chọn giảm

Ta có:
$\int\limits_{-1}^{1}\frac{x^{4} + x^{2}}{x^{2}+1}dx$
$=\int\limits_{-1}^{1}x^2dx$
$=\frac{x^3}{3}\left|\begin{array}{l}1\\-1\end{array}\right.=\frac{2}{3}$