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 2 years ago
I need some help with checking this answer please:
\[\large\int\limits_4^5 \frac{x^33 x^29}{x^33 x^2} dx = x\frac{3}{x}\ln3x+\lnx+C\]
I got this far using the partial fractions technique. I went to check my answer that I got on paper but Wolfram gives some craziness with hyperbolic tangents that I don't understand. So what should the answer be and wouldn't the natural log of 3x be an issue because it gives nonreal values?
The correct answer MUST have real values and be in terms of constants and logs. If somebody could show me what's going on here it would be great :)
 2 years ago
I need some help with checking this answer please: \[\large\int\limits_4^5 \frac{x^33 x^29}{x^33 x^2} dx = x\frac{3}{x}\ln3x+\lnx+C\] I got this far using the partial fractions technique. I went to check my answer that I got on paper but Wolfram gives some craziness with hyperbolic tangents that I don't understand. So what should the answer be and wouldn't the natural log of 3x be an issue because it gives nonreal values? The correct answer MUST have real values and be in terms of constants and logs. If somebody could show me what's going on here it would be great :)

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agentx5
 2 years ago
Best ResponseYou've already chosen the best response.0For my partial fractions I got: \[x^33x^29=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x3}\] The A part turns into lnx, the B part turns into 3/x, the C part turns into ln3x

helder_edwin
 2 years ago
Best ResponseYou've already chosen the best response.1\[ \Large \int\frac{x^33x^29}{x^33x^2}\,dx= \int\,dx\int\frac{9}{x^2(x3)}\,dx \] so you have to solve \[ \Large \frac{9}{x^2(x3)}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x3} \]

Valpey
 2 years ago
Best ResponseYou've already chosen the best response.0Well, for one thing you have written the natural log of the absolute value of 3x.

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.0Yeah I used long division initially to pull out this: \[1 + \frac{9}{x^33x^2}\]

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.0Integral of one with respect to x is just x, logically

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.0I think I mistyped my second post, sry @helder_edwin

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.0@Valpey oh yeah... so it's always a positive...

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.0ln3(4) = ln1 = ln 1 = 0

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.0That's a nice simplification

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.0/me waves hello @asnaseer :) ty for stopping by!

asnaseer
 2 years ago
Best ResponseYou've already chosen the best response.2you didn't need to do long division to simplify this in the first place. you could just have done this:\[\frac{x^33x^29}{x^33x^2}=\frac{x^33x^2}{x^33x^2}\frac{9}{x^33x^2}=1\frac{9}{x^33x^2}\]

asnaseer
 2 years ago
Best ResponseYou've already chosen the best response.2secondly, the logs should not contain the absolute value symbols around them

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.0Good point @asnaseer and why not? Without it wouldn't I have a nonreal answer issue when I go to evaluate?

asnaseer
 2 years ago
Best ResponseYou've already chosen the best response.2you should have done this:\[\int\limits_4^5 \frac{x^33 x^29}{x^33 x^2} dx =[ x\frac{3}{x}\ln(3x)+\ln(x)]_4^5\]\[\qquad=[x\frac{3}{x}+ln(\frac{x}{3x})]_4^5\]\[\qquad=(5\frac{3}{5}+ln(\frac{5}{2}))(4\frac{3}{4}+ln(\frac{4}{1}))\]\[\qquad=5\frac{3}{5}4+\frac{3}{4}+ln(\frac{5}{2}\div\frac{4}{1})\]

asnaseer
 2 years ago
Best ResponseYou've already chosen the best response.2and the minuses cancel out inside the logs

asnaseer
 2 years ago
Best ResponseYou've already chosen the best response.2BTW: waves hello back to @agentx5 :)

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.0Oh... properties of logs... \[\lna\lnb=\ln\frac{a}{b}\]

asnaseer
 2 years ago
Best ResponseYou've already chosen the best response.2yes  but you shouldn't use the '' symbols here

asnaseer
 2 years ago
Best ResponseYou've already chosen the best response.2as they usually indicate absolute value

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.0\[\qquad=5\frac{3}{5}4+\frac{3}{4}+ln(\frac{5}{2}\div\frac{4}{1}) \approx 3.45258\] Disagrees with this check: http://www.wolframalpha.com/input/?i=integrate+from+4+to+5+for+%28x^33x^29%29%2F%28x^33x^2%29 \(\approx \frac{17}{25}\)

asnaseer
 2 years ago
Best ResponseYou've already chosen the best response.2I work it out to be exactly the same as the answer wolfram gives

asnaseer
 2 years ago
Best ResponseYou've already chosen the best response.2I get:\[\frac{23}{20}+ln(\frac{5}{8})\approx 0.68\]

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.0Ugh I see now I used the wrong base log on my calculator... /facepalm

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.0Thank you so much you were a great help! @TuringTest give that claymationman a medal ^_^

asnaseer
 2 years ago
Best ResponseYou've already chosen the best response.2you are more than welcome my friend! :)
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