如何使用分部積分法求出(sin(ax) )^2*dx?

部分積分法： ∫u dv = uv - ∫v du

∫sin²(ax) dx
= ∫sin(ax) [sin(ax) dx]
= -(1/a) ∫sin(ax) d[cos(ax)]
= -(1/a) {sin(ax) cos(ax) - ∫cos(ax) [d sin(ax)]}
= -(1/a) {sin(ax) cos(ax) - a∫cos²(ax) dx}
= -(1/a) {sin(ax) cos(ax) - a∫[1 - sin²(ax)] dx}
= -(1/a) sin(ax) cos(ax) + ∫dx - ∫sin²(ax) dx

2∫sin²(ax) dx = -(1/a) sin(ax) cos(ax) + ∫dx
2∫sin²(ax) dx = -(1/a) sin(ax) cos(ax) + x + C₁

∫sin²(ax) dx = (1/2)[x - (1/a) sin(ax)] + C