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Question

Integrate the function.
x(logx)2dx


Solution

Let I=x(logx)2dx
On taking (logx)2 as first function and x as second function and integrating by parts, we get
I=(logx)2xdx[ddx(logx)2xdx]dx=(logx)2.x22[2logxx.x22]dx=x22(logx)2xlogxdx
Again integrating by parts, we get
I=x22(logx)2[logxxdx{(ddxlogx)xdx}dx]+C=x22(logx)2[x22logx1x.x22dx]+C=x22(logx)2x22logx+12xdx+C=x22(logx)2x22logx+x24+C


Mathematics

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