Question

求x^3/(x+3)的不定积分

Answer 1

x³/3 - (3/2)x² + 9x - 27ln|x + 3| + C

解题过程：

x³/(x + 3)

= x²[(x + 3) - 3]/(x + 3)

= x² - 3x²/(x + 3)

= x² - 3x[(x + 3) - 3]/(x + 3)

= x² - 3x[1 - 3/(x + 3)]

= x² - 3x + 9x/(x + 3)

= x² - 3x + 9[(x + 3) - 3]/(x + 3)

= x² - 3x + 9 - 27/(x + 3),说简单些,就是多项式除法化简而已,不过这里不方便写.

∫ x³/(x + 3) dx

= ∫ x² dx - 3∫ x dx + 9∫ dx - 27∫ dx/(x + 3)

= x³/3 - (3/2)x² + 9x - 27ln|x + 3| + C

扩展资料：

不定积分的公式

1、∫ a dx = ax + C，a和C都是常数

2、∫ x^a dx = [x^(a + 1)]/(a + 1) + C，其中a为常数且 a ≠ -1

3、∫ 1/x dx = ln|x| + C

4、∫ a^x dx = (1/lna)a^x + C，其中a > 0 且 a ≠ 1

5、∫ e^x dx = e^x + C

6、∫ cosx dx = sinx + C

7、∫ sinx dx = - cosx + C

Answer 2

∫ x³ / (x + 3) dx
= ∫ (x³ + 27 - 27) / (x + 3) dx
= ∫ (x³ + 27) / (x + 3) dx - 27 ∫ dx / (x + 3)
= ∫ (x + 3) (x² - 3 x + 9) / (x + 3) dx - 27 ∫ dx / (x + 3)
= ∫ (x² - 3 x + 9) dx - 27 ∫ d(x + 3) / (x + 3)
= x³ / 3 - 3 x² / 2 + 9 x - 27 ln|x + 3| + C

Answer 3

x^3/(x+3)
=(x^3+27)/(x+3)-27/(x+3)
=x^2-3x+9-27/(x+3)