不定積分∫x^a(lnx)^2dx的計算

主要内容：

通過多次分部積分法介紹不定積分∫x^a(lnx)^2dx的計算步驟。

通式推導：

∫x^a(lnx)^2dx

=1/(a 1)∫(lnx)^2dx^(a 1)，以下第一次使用分部積分法，

=1/(a 1) (lnx)^2*x^(a 1) -1/(a 1)∫x^(a 1) d(lnx)^2

=1/(a 1) (lnx)^2*x^(a 1) -2/(a 1)∫x^(a 1) *lnx*(1/x)dx

=1/(a 1) (lnx)^2*x^(a 1) -2/(a 1)∫x^a*lnxdx

=1/(a 1) (lnx)^2*x^(a 1) -2/(a 1)^2∫lnxdx^a11,以下第二次使用分部積分法，

=1/(a 1) (lnx)^2*x^(a 1) -2/(a 1)^2lnx*x^(a 1) 2/(a 1)^2∫x^(a 1) dlnx

=1/(a 1) (lnx)^2*x^(a 1) -2/(a 1)^2lnx*x^(a 1) 2/(a 1)^2∫x^(a 1) *1/xdx

=1/(a 1) (lnx)^2*x^(a 1) -2/(a 1)^2lnx*x^(a 1) 2/(a 1)^2∫x^adx

=1/(a 1) (lnx)^2*x^(a 1) -2/(a 1)^2lnx*x^(a 1) 2/(a 1)^3x^(a 1) c

=x^(a 1) [(lnx)^2/(a 1) -2/(a 1)^2lnx 2/(a 1)^3] c

舉例計算，例如當a=4時，計算過程如下：

∫x^4 (lnx)^2dx

=(1/5)∫(lnx)^2dx^a11，以下第一次使用分部積分法，

=(1/5) (lnx)^2*x^5-(1/5)∫x^5d(lnx)^2

=(1/5) (lnx)^2*x^5-(2/5)∫x^5*lnx*(1/x)dx

=(1/5) (lnx)^2*x^5-(2/5)∫x^4*lnxdx

=(1/5) (lnx)^2*x^5-(2/25)∫lnxdx^5,以下第二次使用分部積分法，

=(1/5) (lnx)^2*x^5-(2/25)lnx*x^5 (2/25)∫x^5dlnx

=(1/5) (lnx)^2*x^5-(2/25)lnx*x^5 (2/25)∫x^5*1/xdx

=(1/5) (lnx)^2*x^5-(2/25)lnx*x^5 (2/25)∫x^adx

=(1/5) (lnx)^2*x^5-(2/25)lnx*x^5 (2/125)x^5 c

=x^5 [(1/5) (lnx)^2-(2/25)lnx (2/125)] c

=(1/125)x^5 [25 (lnx)^2-10lnx 2] c.

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