# prove the following reduction formula: `int (lnx)^ndx=x(lnx)^n-n int (lnx)^{n-1}dx`

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Expert Answers

lfryerda | Certified Educator

To prove the reduction formula `int (lnx)^ndx=x(lnx)^n-nint(lnx)^{n-1}dx` we need to use the integration by parts formula `int udv=uv-int vdu` .

For this formula, let `u=(lnx)^n` and `dv=dx` , so `du=n/x(lnx)^{n-1}dx` and `v=x` . This means that

`int (ln x)^ndx`

`=x(lnx)^n-int n/x(lnx)^{n-1} xdx`

`=x(lnx)^n-nint(lnx)^{n-1}dx`

**The reduction formula has been proved.**