Question

∫（3x²×x²+2x²)/(x²+1)dx

Answer 1

∫（3x²×x²+2x²)/(x²+1)dx =∫（3x²×x²+6x²+3)/(x²+1)dx-∫(4x²+4)/(x²+1)dx +∫ 1/(x²+1)dx
=∫3(x²+1)dx -∫4dx+∫ 1/(x²+1)dx=x^3 -x + arctanx＋C',C'常数
其中∫ 1/(x²+1)dx=arctanx＋C是公式。

Answer 2

∫（3x²×x²+2x²)/(x²+1)dx
=∫x²(3x²+3-1)/(x²+1)dx
=∫x²(3-1/(x²+1))dx
=∫[3x²-x²/(x²+1)]dx
=∫[3x²-(x²+1-1)/(x²+1)]dx
=∫[3x²-1+1/(x²+1)]dx
=3∫x²dx-∫1dx+∫1/(x²+1)]dx
=x³-x+arctanx+C
C'常数