# Thread: Flawed Knowledge of Reduction Formulas

1. ## Flawed Knowledge of Reduction Formulas

Any explanation of how to evaluate this problem through the use of reduction formulas would be greatly appreciated.

Evaluate: $\int(x^nln(x)dx)$ where $x\not=-1$

I know that one must use integration by parts ( $\int(udv)=uv-\int vdu$)
As it stands right now I have $u=x^n$ $\rightarrow du=nx^{n-1}dx$
Along with $dv=ln(x)dx$ $\rightarrow v=xln(x)-x+C$

I substitute the above values into the equation for integration by parts and I do not know how to continue from there. Any advice would be appreciated.

2. Originally Posted by LithiumPython
Any explanation of how to evaluate this problem through the use of reduction formulas would be greatly appreciated.

Evaluate: $\int(x^nln(x)dx)$ where $x\not=-1$

I know that one must use integration by parts ( $\int(udv)=uv-\int vdu$)
As it stands right now I have $u=x^n$ $\rightarrow du=nx^{n-1}dx$
Along with $dv=ln(x)dx$ $\rightarrow v=xln(x)-x+C$

I substitute the above values into the equation for integration by parts and I do not know how to continue from there. Any advice would be appreciated.

It looks pretty straightforward and what you did is right (without the integration constant, of course) , so:

$\int x^n\ln x\,dx= x^{n+1}\ln x - x^{n+1}-n\!\!\int x^n\ln x\,dx+n\!\!\int x^n\,dx\,\Longrightarrow$ $(n+1)\!\!\int x^n\ln
x\,dx=x^{n+1}\left(\ln x-1+\frac{n}{n+1}\right) + C$
...

Tonio