Question

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

Answer 1

∫（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

Answer 2

∫[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]
=π/2-√3/2+π/6
=2π/3-√3/2

Answer 3

令 t=arctan x, x=tan t ,dx=dtan t,原式=