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Use integration by parts to prove the reduction formula.$\int(\ln x)^{n} d x=x(\ln x)^{n}-n \int(\ln x)^{n-1} d x$

$\int(\ln x)^{n} d x=n(\ln x)^{n}-n \int(\ln x)^{n-1} d x$

Calculus 1 / AB

Calculus 2 / BC

Chapter 5

Integrals

Section 5

Integration by Parts

Integration Techniques

Missouri State University

Oregon State University

Boston College

Lectures

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Hello and welcome. We are looking at Chapter five, Section five Problem 29. Rest of prove a reduction formula for natural log. That's what we'll do. All right, So Ah, here's our natural log. We're going to use integration by parts. So if we just choose you to be everything in that Oh integral Devi to be, uh, just what's left behind. And when we take the derivative of you we have and my ass one, we're sorry just and I was getting ahead of myself End times natural log of X and that's raised to the N minus one. Then that's multiplied by the derivative of natural log, which is one over X and then the Of course we take the anti derivative. There's just gonna be X. Don't forget my d x here. That's all the pieces we need. You could just apply the formula. That's you ve so this would be I'll do the you So it's easier to read. X natural log of X, the end minus the integral of VD. You so the ex seeing you, what is that longer part there? Now, if I simplify this, I have a one over X here that's gonna cancel. With this one over X, they're gonna multiply and become one. This is gonna be minus. Um Oh, and one more thing I can do is I can take this and outside the integral, because it's just a constant. So it's gonna be minus and times the integral of us of these to cancel. And I sent that one away. I'm left with natural log of X. Well, the natural order of X, the n minus one D x. All right. And so that is exactly what our reduction formula says. Um, so this right here the process is the answer for a proof based problem like this, But I got to the right ending point. So? So we're good. We're done.

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