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Evaluate the integrals that converge.$$\int_{-\infty}^{0} \frac{d x}{(2 x-1)^{3}}$$

$-\frac{1}{4}$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

PRINCIPLES OF INTEGRAL EVALUATION

Section 8

Improper Integrals

Integrals

Integration

Integration Techniques

Trig Integrals

Trig Substitution

Campbell University

University of Michigan - Ann Arbor

Boston College

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Okay, so here we have. Well, the integral going from native infinity, Um, zero off DX over two X minus one cube. So we just go where to go ahead and replace native infinity with what would be with a or that I'm using. Be here. So I'm taking the limit now. As be, um, tens to Well, it tends to native infinity. Right off the inner girl going from B zero now off the X over the quantity two X minus one pupil. Okay, so you go ahead and we're gonna evaluate. Well, the indefinite integral DX over two X minus one cube. We're gonna use a u substitution, and I'm gonna let you be equal to two AKs minus one. They were You was equal to X minus one. Um, that implies that two D X is equal to do you, which implies that the X is equal to one half. Do you? Okay, now we have, or the integral of DX over two X minus one. Cube is equal to one half times the integral off. Just, um, won over you, Cube, is you to the minus three. Do you? Okay, This is equal to one half times while you to the minus two divided my by minus two, which is going to be equal to this becomes a minus one over for you squared. And, of course, while classy. Right, Um, which is equal to then back substituting, um, for you. What is you, Will you was equal to two X minus one. This is equal to negative 1/4 times you, But you is two x minus one. Um, squared. Okay. And then, of course, wealth, of course. Plus C right for the indefinite, Integral. Okay, there's no way a man for us because, um, we're taking while having a limit here to take the limit. Now, remember as well as be, um I was gonna go to infinity. Oh, well, of, um, negative 1/4 times two x minus one square. And now we're evaluating from B to zero. So, um, this is gonna be equal to well, again. We have. Of course, the limit as be tends to infinity. Oh, well, we get a negative 1/4 times zero minus one square, which is becomes a minus 1/4 and then Mary plus 1/4 times to B minus. one birth. So we have plus 1/4 times to be minus one quality squared. Okay, Now we go ahead and we take the limit. This is equal to well, negative. 1/4. There's no lipstick there. Then we have while plus the limit as bi coastal infinity of 1/4 times B minus one squared. But as, um as b goes to infinity, Right. The denominator here isn't Goldman. Infinity times. Infinity minus one. Infinity squared. Infinity times for infinity. We have one over infinity, which is just zero. This is native 1/4 plus zero, which is equal to negative 1/4. So the integral here converges and it converges to Negative. Juan fourth. All right, take care. Are you

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