Why Does the Integral of 1/(sqrt(u^2-4)) Lead to a Natural Log Function?

[Jules18]

\int\stackrel{du}{\sqrt{u^{2}-4}} = ln(u + sqrt.(u²-4))

My answer key makes the jump directly from that integral to that natural log fxn, which would make sense to me if only the "u +" weren't inside the brackets of the ln fxn.
Does anyone know where it comes from?

 

[Jules18]

I'm sorry about the latex. I edited it but the changes aren't showing up. Hope you can still decipher it.
The latex program is ridiculously glitchy, but I'm sure I'm not the first person to say that.

 

[snipez90]

Um, that's the correct antiderivative? Have you differentiated it to check? Perhaps you forgot the chain rule.

 

[ƒ(x)]

Try trig substitution

 

[hamster143]

If u = \cosh{t}, then u^2-1 = \sinh^2{t}, and du = \sinh{t} dt.

 

[HallsofIvy]

\int\stackrel{du}{\sqrt{u^{2}-4}} = ln(u + sqrt.(u²-4))

My answer key makes the jump directly from that integral to that natural log fxn, which would make sense to me if only the "u +" weren't inside the brackets of the ln fxn.
Does anyone know where it comes from?

Actually, that shouldn't make sense.
\int \frac{1}{f(x)}dx
is NOT, in general, equal to ln(f(x))!

hamster143 has suggested you use u= cosh(t) because cosh^2(t)- 1= sinh^2(t) and you can get rid of the square root.

It is also true that sec^2(t)- 1= csc^2(t) so the substitution u= sec(t) would also work. It might be a little more complicated because then du= sec(t)tan(t)dt.

(This board has a slight problem with its LaTex. When you edit LaTex, always click the "refresh" button on your screen (looks like two little 'circulating' green arrows). That should fix it.)

 

[Jules18]

Actually, that shouldn't make sense.
\int \frac{1}{f(x)}dx
is NOT, in general, equal to ln(f(x))!

hamster143 has suggested you use u= cosh(t) because cosh^2(t)- 1= sinh^2(t) and you can get rid of the square root.

It is also true that sec^2(t)- 1= csc^2(t) so the substitution u= sec(t) would also work. It might be a little more complicated because then du= sec(t)tan(t)dt.

(This board has a slight problem with its LaTex. When you edit LaTex, always click the "refresh" button on your screen (looks like two little 'circulating' green arrows). That should fix it.)

Really? the equality I posted is straight from my prof's answer key. ...

Also, I'm not sure I understand what you mean by cosh^2(t). Is that another way of expressing cos^2(ht)?

In that case I thought cos^2(x) + sin^2(x) = 1,
so how could cos^2(x) - 1 = sin^2(x) ? I was thinking it would equal -sin^2(x) , and so I was getting confused because it's under a sqrt. sign so it can't be -ve, so I resorted to the answer key only to find my prof's weird equality ... -_-

(thanks for the LaTex tip)

 

[hamster143]

Cosh and sinh are hyperbolic sine and cosine, defined as

\sinh{t} = (e^t-e^{-t})/2
\cosh{t} = (e^t+e^{-t})/2

 

[hover]

Ya your answer isn't right. Remember that when integrating, derivatives are your best friend to check your work. If I were to take a derivative of that I would get a u + sqrt.(u2-4) on the bottom * whatever the derivative of that is on the top.

You need to use a trig substitution here of some sort.

 

[tt2348]

woahhhh no youre answer is definitely right.

\frac{d}{dx}(ln(x+\sqrt{x^{2}-4})=\frac{1}{x+\sqrt{x^{2}-4}}*(1+\frac{x}{\sqrt{x^{2}-4}})=\frac{1}{x+\sqrt{x^{2}-4}}*\frac{x+\sqrt{x^{2}-4}}{\sqrt{x^{2}-4}}=\frac{1}{\sqrt{x^{2}-4}}

 

[tt2348]

use x=2*sec(u) and dx=2*sec(u)*tan(u)

so the problem would become something like \int \frac{2*\sec u *\tan u}{2*\sqrt{(\sec x)^{2}-1}}du

you'll get \int \sec u