x^5比上x-1的不定积分

Question

x^5比上x-1的不定积分

Answer 1

x⁵
= x⁴(x - 1) + x⁴
= x⁴(x - 1) + x³(x - 1) + x³
= x⁴(x - 1) + x³(x - 1) + x²(x - 1) + x²
= x⁴(x - 1) + x³(x - 1) + x²(x - 1) + x(x - 1) + (x - 1) + 1
∫ x⁵/(x - 1) dx
= ∫ [x⁴(x - 1) + x³(x - 1) + x²(x - 1) + x(x - 1) + (x - 1) + 1]/(x - 1) dx
= ∫ (x⁴ + x³ + x² + x + 1) dx + ∫ dx/(x - 1)
= x⁵/5 + x⁴/4 + x³/3 + x²/2 + x + ln|x - 1| + C