anonymous
  • anonymous
How do I integrate this?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
|dw:1326789204685:dw|
anonymous
  • anonymous
Is this how to do it? |dw:1326789228310:dw|
anonymous
  • anonymous
No, your last step is incorrect (try taking the derivative of your final expression to see why)

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anonymous
  • anonymous
Instead, you need to use the identity \[\int \frac{dx}{x^2+a^2}=\tan^{-1}\frac{x}{a}+C\] Take the derivative of the right expression to see why this works!
anonymous
  • anonymous
you'll need some trigonometry to make the u-substitution u = tan(x)/5
ash2326
  • ash2326
no , \[\int\limits_{}^{} 40/( x^{2}+25) dx\] \[\int\limits_{}^{} 40/(x^2+5^2) dx\] \[40 \tan^{-1} (x/5) +c\]
anonymous
  • anonymous
How can I work with that? |dw:1326789616311:dw|
anonymous
  • anonymous
By the chain rule: \[\frac{d}{dx}\tan^{-1}\frac{x}{a}=\frac{1}{1+(\frac{x}{a})^2} \cdot \frac{1}{a}\] Simplify a bit and you will get your answer.
anonymous
  • anonymous
Sorry, the identity is \[\int \frac{a \cdot dx}{x^2+a^2}=\tan^{-1}\frac{x}{a}+C\] My bad.
anonymous
  • anonymous
Thus, for the sake of completeness, the answer is \[\int \frac{40dx}{x^2+25}=8\tan^{-1}\frac{x}{5}+C\]
anonymous
  • anonymous
Do you multiply 40 by (1/5) because of the chain rule?
anonymous
  • anonymous
Mhm, that you do!

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