I am stuck on this integral problem

[tulip kestrel]

Can you find any similarities between the linear expression in the numerator and the quadratic expression in the denominator?

Sure, that's definitely a good option, but it doesn't seem to work in this case. Is there a way we can rewrite the term in the denominator in terms of x-2?

You're good. The square root's really annoying here, but if we can turn just the quadratic into something nicer, we might be able to work with it down the line.

The best plan of attack in this case is probably going to be u-substitution.

In a perfect world, we'd like our u to be x-2, but we can't factor x^2-4x+3 into x-2.

Yes. since x-2 doesn't work nicely, is there another u which seems natural?

That's cool. It's a pretty tough u-sub. We've tried the numerator, what happens when we try the denominator? Setting u to equal the whole denominator is a bit complicated, so is there another expression whose derivative might be a bit simpler?

We tried $u=x-2$ and it was too simple and we tried $u=\sqrt{x^{2}-4x+3}$ and it was too complicated. Is there a $u$ value which fits somewhere inbetween in terms of complexity?

[sour spear]

CFM*1034

[tulip kestrel]

Our first u is a linear expression, while our second u is a quadratic inside of a square root. If we want something similar to our second u but without both the square root and the quadratic expression, what could we do?

That's a great idea. What happens when you do that?

So you have that du/dx = 2x-4. What happens when you substitute du for dx in the integral given this equation?

Then this is definitely an interesting problem. In this case, we have that du/dx = 2x - 4, so dx = du/(2x-4). Then, our integral becomes $\int\limits_{x=1}^{x=4} \frac{x-2}{\sqrt{u}}\cdot\frac{du}{2x-4}$. We can then cancel out terms to get that this integral equals $\int\limits_{x=1}^{x=4} \frac{1}{2\sqrt{u}} du$, which you might've seen before. From there, we get that the expression equals $\sqrt{u}$ at $x=4$ $- \sqrt{u}$ at $x=1$.

Then, we can substitute our expression for u back into this formula to get that this is equal to $\sqrt{4^{2}-4\cdot 4 +3}-\sqrt{1^{2}-4\cdot 1 +3} = \sqrt{3}$.

[sour spear]

CFM*1034