求不定积分∫x^2*arctanxd(arctanx)

Question

求不定积分∫x^2*arctanx d(arctanx)

Answer 1

∫ x²arctanx d(arctanx)、先将1/(1 + x²)凑成d(arctanx)这步尝试过是行不通的、拉出来重新组织.
= ∫ (x²arctanx)/(1 + x²) dx
= ∫ [(1 + x²) - 1]arctanx/(1 + x²) dx
= ∫ arctanx dx - ∫ arctanx/(1 + x²) dx
= xarctanx - ∫ x d(arctanx) - ∫ arctanx d(arctanx)、这里可以拉进去
= xarctanx - ∫ x/(1 + x²) dx - (1/2)(arctanx)²
= xarctanx - (1/2)ln(1 + x²) - (1/2)(arctanx)² + C