Question

求x^3/(x+3)的不定积分时的思路怎么走

Answer 1

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