Giải thích các bước giải:
Bài 1:
$I=\int \dfrac{\ln^3x}{x^2}dx$
$=\int \ln^3xd\dfrac{-1}{x}$
$=-\dfrac{\ln^3x}{x}-\int \dfrac{-1}{x}d(\ln^3x)$
$=-\dfrac{\ln^3x}{x}+\int \dfrac{3\ln^2x}{x^2}dx$
$=-\dfrac{\ln^3x}{x}+\int \dfrac{3\ln^2x}{x^2}dx$
Tiếp tục tích phân từng phần
$\to I=-\dfrac{\ln^3x}{x}+3(-\dfrac{\ln^2x}{x}+2(-\dfrac{\ln x}{x}-\dfrac 1x))+C$
Bài 2:
$\int x(\arctan x)^2dx$
$=\dfrac 12\int (\arctan x)^2dx^2$
$=\dfrac 12(x^2\arctan x-\int x^2d(\arctan x)^2)$
$=\dfrac 12(x^2\arctan x-\int x^2.\dfrac{2}{x^2+1}.\arctan x dx$
$=\dfrac 12.x^2\arctan x-\int \dfrac{x^2}{x^2+1}.\arctan x dx$
$=\dfrac 12.x^2\arctan x-\int (\arctan x+\dfrac{1}{x^2+1}.\arctan x) dx$
$=\dfrac 12x^2.\arctan^2x-x\arctan x+\dfrac 12\arctan^2x+\ln\sqrt{x^2+1}+C$
