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thomas5267
 2 years ago
Rewrite \(\int^3_2\frac{3x}{x^2+4x+5}\,dx\) into the form of \(\int^b_a\frac{mt+c}{t^2+k^2}\) or \(\int^b_a\frac{mt+c}{t^2k^2}\).
Not even a clue how to do it. Please help.
thomas5267
 2 years ago
Rewrite \(\int^3_2\frac{3x}{x^2+4x+5}\,dx\) into the form of \(\int^b_a\frac{mt+c}{t^2+k^2}\) or \(\int^b_a\frac{mt+c}{t^2k^2}\). Not even a clue how to do it. Please help.

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Callisto
 2 years ago
Best ResponseYou've already chosen the best response.3Consider the denominator: \[x^2+4x+5=(x^2 +4x+44)+5 = (x+2)^2 4+5 =...\]

thomas5267
 2 years ago
Best ResponseYou've already chosen the best response.0I don't think so. Completing the square will yield \((x+a)^2+b\), which is not what we want...

Callisto
 2 years ago
Best ResponseYou've already chosen the best response.3Hmm... \[\int^3_2\frac{3x}{x^2+4x+5} dx \]\[= \int^3_2\frac{3x}{x^2+4x+44+5} dx\]\[=\int^3_2\frac{3x}{(x+2)^24+5} dx\]\[=\int^3_2\frac{3x}{(x+2)^2+1} dx\]So, in this case, t= x+2, k=1 For the numerator, we want mt+c, and we have 3x. m must be 3. 3(x+2) +c = 3x c = 6

Callisto
 2 years ago
Best ResponseYou've already chosen the best response.3Ha! Maybe I'm not even going in the right direction!

thomas5267
 2 years ago
Best ResponseYou've already chosen the best response.0OK, but then the problem tells us to integrate it. \[ \begin{align*} &\int^3_2\frac{3x}{(x+2)^2+1}\,dx\\ =&\int^5_3\frac{3t6}{t^2+1}\,dt \end{align*} \] As far as I can see, this cannot be integrated, or is my brain not functioning properly now.

Callisto
 2 years ago
Best ResponseYou've already chosen the best response.3When x=2, t= 2+2 = 4... \[\int_4^5\frac{3t6}{t^2+1}dt\]\[=\int_4^5\frac{3t}{t^2+1}\frac{6}{t^2+1}dt\] For \(\int\frac{3t}{t^2+1}dt\) \[\int\frac{3t}{t^2+1}dt = \frac{3}{2}\int\frac{1}{t^2+1}d(t^2) \]Shouldn't be difficult For\(\int\frac{6}{t^2+1}dt\), use trigo sub.

thomas5267
 2 years ago
Best ResponseYou've already chosen the best response.0OK, my brain is certainly not working properly. What is happening in here? I have never see something like this. \[ \int\frac{3t}{t^2+1}\,dt=\frac{3}{2}\int\frac{1}{t^2+1}\,d(t^2) \]

Callisto
 2 years ago
Best ResponseYou've already chosen the best response.3d/dt (t^2) = 2t, agree?

thomas5267
 2 years ago
Best ResponseYou've already chosen the best response.0I still don't understand \(d(t^2)\) and what should I do with it. I would do this in the following way. \[ \begin{align*} &\int\frac{3t}{t^2+1}\,dt\\ =&3\int\frac{t}{t^2+1}\,dt\\ =&\frac{3}{2}\int\frac{1}{u}\,du\\ =&\frac{3}{2}\ln(t^2+1)+C \end{align*} \]

Callisto
 2 years ago
Best ResponseYou've already chosen the best response.3Basically, it's just substitution.. u = t^2 +1 du = 2t dt t dt = du/2 But you keep it as t, anyway, forget it! And you get it right.

Callisto
 2 years ago
Best ResponseYou've already chosen the best response.3But remember it's definite integral.. not indefinite..

thomas5267
 2 years ago
Best ResponseYou've already chosen the best response.0Can you teach me how to do this in your way? It seems way faster then mine.

Callisto
 2 years ago
Best ResponseYou've already chosen the best response.3Hmm... I just did integration for the numerator and put it after ''d'' \[\int\frac{3t}{t^2+1}dt\]If you integrate the numerator, you'll get \(\frac{t^2}{2}\)So, \[\int\frac{3t}{t^2+1}dt = 3\int\frac{t}{t^2+1}dt =3 \int\frac{1}{t^2+1}d(\frac{t^2}{2})=\frac{3}{2} \int\frac{1}{t^2+1}d(t^2) \] Another way to think about it is that, d/dt (t^2 /2) gives you t, the numerator

Callisto
 2 years ago
Best ResponseYou've already chosen the best response.3Did... I make it worse??

thomas5267
 2 years ago
Best ResponseYou've already chosen the best response.0So I can just integrate the numerator? What should I do with \(\int\frac{1}{t^2+1}d(t^2)\) then?

Callisto
 2 years ago
Best ResponseYou've already chosen the best response.3Get the answer! lnt^2 +1 +C! Just consider t^2 = u, it's simply \(\int \frac{1}{u+1}du\) , that is ln u+1+C = lnt^2+1+C

thomas5267
 2 years ago
Best ResponseYou've already chosen the best response.0Is the following true? \[ \int^5_4\frac{3t}{t^2+1}dt = \frac{3}{2}\int^{25}_{16}\frac{1}{t^2+1}d(t^2) \]

Callisto
 2 years ago
Best ResponseYou've already chosen the best response.3No, you didn't change the variable, so you don't have to change the bound value.

thomas5267
 2 years ago
Best ResponseYou've already chosen the best response.0What is the name of the procedure? I would like to find more information on the internet.

Callisto
 2 years ago
Best ResponseYou've already chosen the best response.3I'm sorry.. I don't know the name of it :(

thomas5267
 2 years ago
Best ResponseYou've already chosen the best response.0Anyways, thank you! I have learnt a lot in here.
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