计算：定积分∫（在上 √3,在下0 ）xarctan xdx求详细过程答案，拜托大神

显示全部楼层 · 2012-12-19 14:39:55

∫（在上 √3,在下0 ）xarctan xdx=∫（在上 √3,在下0 ）arctan xdx2/2=x2/2arctanx|(0->√3)-1/2∫(0->√3)x2/(1+x2)dx=π/2 -1/2 ∫（0->√3）（1-1/(1+x2)）dx=π/2-√3/2+1/2arctanx|(0->√3)=π/2-√3/2+π/6=2π/3-√3/2...

千问 · 2012-12-19 14:39:55

∫[0,√3]xarctan xdx=1/2∫[0,√3]arctan xdx^2=1/2x^2arctanx[0,√3]-1/2∫[0,√3]x^2darctan x=π/2-1/2∫[0,√3]x^2/(1+x^2)dx=π/2-1/2∫[0,√3][1-1/(1+x^2)]dx=π/2-1/2(x-arctanx)[0,√3...

千问 · 2012-12-19 14:39:55

令 t=arctan x, x=tan t ,dx=dtan t,原式=...