Question

这道微积分题怎么做?(分部积分法)

麻烦给出详细解答 还有说一下做这些题的技巧啥的，谢谢了！！！

Answer 1

方法如下，
请作参考：

Answer 2

分部积分法

Answer 3

换一种思路：由微分求积分
[(1/3)x^3 arctan(3x)]'
= x^2 arctan(3x) + x^3/(1+9x^2)
= x^2 arctan(3x) + (1/9)x - (x/9)/(1+9x^2)
Therefore,
∫x^2 arctan(3x) dx = (1/3)x^3 arctan(3x) - x^2/18 + (1/162)ln(1+9x^2) + c

Answer 4

∫x^2 arctan3x dx = (1/3)∫arctan3x dx^3
= (1/3) x^3 arctan3x - (1/3)∫[3x^3/(1+9x^2)] dx
= (1/3) x^3 arctan3x - (1/9)∫[(9x^3+x-x)/(1+9x^2)] dx
= (1/3) x^3 arctan3x - (1/9)∫[x-x/(1+9x^2)] dx
= (1/3) x^3 arctan3x - (1/18)x^2 + (1/9)∫[x/(1+9x^2)] dx
= (1/3) x^3 arctan3x - (1/18)x^2 + (1/162)ln(1+9x^2) + C