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anonymous

  • 5 years ago

Use an appropriate change of variable to evaluate the convergent improper integral 1/[(x^0.5)(1+x)] what should i use?

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  1. anonymous
    • 5 years ago
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    the answer is (pi) but i dunno how should i get to the answer

  2. anonymous
    • 5 years ago
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    Is it dy/dx = 1/[(x^0.5)(1+x)] ? Move the dx to the right side, integrate the left and you get "y", integrate the right side now and try using a trig substitution...

  3. anonymous
    • 5 years ago
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    sorry..my bad... 1st time use this website...but i found this fun ^ ^ the question is based on improper integral. \[\int\limits\limits_{0}^{\infty} 1\div [ \sqrt{x} (1+x) ]\]

  4. anonymous
    • 5 years ago
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    Okay, for the right hand side, you're going to use a u-substitution (try u = sqrt(x)) and after the sub, you should see a pretty clean inverse trig function that pops out.

  5. anonymous
    • 5 years ago
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    Then, finding evaluating at the limits will be an easy task. :)

  6. anonymous
    • 5 years ago
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    *eliminate "finding" :P typo.

  7. anonymous
    • 5 years ago
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    thanks for helping but i cant get through with the trig part.. i cant see which i use to substitute i stuck at \[1\div (u + u ^{3})\] :(

  8. anonymous
    • 5 years ago
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    I've been scratching my head at this one for a while, but I just got it. :) You've got du=1/(2sqrt(x)) dx, right? You've gotta multiply the left by 2sqrt(x) to get dx = 2√x du! Plug THAT into your integral and you'll get 2 * ∫ 1/(1+u^2) du. That should be much easier to solve!

  9. anonymous
    • 5 years ago
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    hm...i getting 0 at the end .. hm... odd.. the answer i getting from my lecture is Pi what is the answer u getting? i get in the end \[\tan^{-1} u \]

  10. anonymous
    • 5 years ago
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    Exactly, you're just forgetting to multiply by 2. So, your final expression in the end is 2 * arctan√x from 0 to ∞. The value of arctan(√x) as x approaches infinity is π/2, and as x approaches 0, is 0! So you get 2* (π/2-0) after applying the limits and the final value is π.

  11. anonymous
    • 5 years ago
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    damn i keep doing so stupid mistake...i might need my brain rest for a while..XD..it's been 5 hours for me ... thanks..u r awesome.. ^ ^

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