tiffany_rhodes
  • tiffany_rhodes
How to find the antiderivative of:
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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tiffany_rhodes
  • tiffany_rhodes
\[\int\limits_{1}^{x}\sqrt{1+(t^2-(1/(4t^2))^2} dt\]
tiffany_rhodes
  • tiffany_rhodes
I'm sorry it's (t^2 -(1/(4t^2)))^2
zepdrix
  • zepdrix
\[\large \int\limits_1^x \sqrt{1+\left(t^2-\frac{1}{4t^2}\right)^2}\] It's a little hard to read the way you've written it out. Is this correct?

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zepdrix
  • zepdrix
Woops I forgot my dt :D whatev.
tiffany_rhodes
  • tiffany_rhodes
lol yes! Sorry for the confusion :)
zepdrix
  • zepdrix
Mmk so let's work from the inside out.\[\large t^2-\frac{1}{4t^2}\qquad = \qquad \frac{4t^4-1}{4t^2}\]I got a common denominator and combined them.
zepdrix
  • zepdrix
Now we have to square this..? Hmm
zepdrix
  • zepdrix
Squaring that term gives us this, \[\large \frac{16t^8-8t^4+1}{16t^4}\] Any confusion on how I got that?
tiffany_rhodes
  • tiffany_rhodes
nope, I got it!
zepdrix
  • zepdrix
So I guess that leaves us here, \[\large \int\limits\limits_1^x \sqrt{1+\frac{16t^8-8t^4+1}{16t^4}}\quad dt\] Once again, we'll want to get a common denominator between that 1 and the fraction.
zepdrix
  • zepdrix
\[\large \int\limits\limits\limits_1^x \sqrt{\frac{16t^4}{16t^4}+\frac{16t^8-8t^4+1}{16t^4}}\quad dt\]
zepdrix
  • zepdrix
\[\large \int\limits\limits\limits\limits_1^x \sqrt{\frac{16t^8+8t^4+1}{16t^4}}\quad dt\]
tiffany_rhodes
  • tiffany_rhodes
hmmm okay. Now I just need to find the antidertivative of that?
zepdrix
  • zepdrix
I have a feeling what we're going to have to do is probably... complete the square in the numerator, and then apply a trig sub. Have you done work with trigonometric substitutions?
zepdrix
  • zepdrix
We can take the root of the denominator really nicely. Let's do that, giving us,\[\large \int\limits_1^x \frac{1}{4t^2}\sqrt{16t^8+8t^4+1}\quad dt\]
tiffany_rhodes
  • tiffany_rhodes
No I haven't. I think we are skipping that section.
zepdrix
  • zepdrix
Ah ok, maybe that's not the next step then. Hmm what to do next. Thinkinggg
Zarkon
  • Zarkon
factor
Zarkon
  • Zarkon
under the radical
Zarkon
  • Zarkon
\[(4t^4+1)^2\]
zepdrix
  • zepdrix
Oh it's already a perfect square under the radical? Blah, I dunno why I didn't notice that lol.
zepdrix
  • zepdrix
\[\large \int\limits\limits_1^x \frac{1}{4t^2}\sqrt{(4t^2+1)^2}\quad dt\]Hmm that works out nicely :)
zepdrix
  • zepdrix
Do you think you can solve it from here tiffany? :D
tiffany_rhodes
  • tiffany_rhodes
yes, thank you both! :)

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