∫x^2/(a^6-x^6)dx(a＞0)

定积分求解

∫x^2/(a^6-x^6) dx=1/3∫1/(a^6-x^6) dx^3=1/(6a^3)∫[1/(a^3-x^3)+ 1/(a^3+x^3)]dx^3
=1/(6a^3)*ln\(a^3-x^3)(a^3+x^3) \+c

∫ x²/(a² - x⁶) dx = ∫ 1/[a² - (x³)²] d(x³/3) = (1/3)∫ 1/[(a - x³)(a + x³)] d(x³) = (1/3)[1/(2a)]∫ [(a + x³) + (a - x³)]/[(a - x³)(a + x³)] d(x³) = [1/(6a)]∫ [1/(a - x³) + 1/(a + x³)] d(x³) = [1/(6a)] * [- ln|a - x³| + ln|a + x³|] + c = [1/(6a)]ln|(a + x³)/(a - x³)| + c