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Ask iit jee aieee pet cbse icse state board community Community Discussion Question: Integrals challenge 1
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[Post New]11 Mar 2008 10:10:17 IST
    Subject: Integrals challenge 1
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elastiboysai (2327)

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[Post New]11 Mar 2008 10:49:30 IST
    Subject: Re:Integrals challenge 1
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konichiwa2x (2224)

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Is this the answer  ?
Guide to latex:
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[Post New]11 Mar 2008 10:59:42 IST
    Subject: Re:Integrals challenge 1
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konichiwa2x (2224)

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=
 
Guide to latex:
http://www.goiit.com/posts/list/community-shelf-a-guide-to-latex-48056.htm

JEE and OLYMPIA INFINATUM
http://iit-redefined.theforum.name/index.php
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[Post New]11 Mar 2008 11:04:52 IST
    Subject: Integrals challenge 1
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Radon222 (161)

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Koniciwa2x how have u managed to decide that this particular substitution for t will work in this question ?
Caution: Radioactive Hazard
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[Post New]11 Mar 2008 11:15:07 IST
    Subject: Re:Integrals challenge 1
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elastiboysai (2327)

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Beautiful, Koni
almost d same as my soln..
 
Here's the basic idea with such problems.
 
Take a look at the img below
downld if necessary
 
here's something about Euler's substitution

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[Post New]11 Mar 2008 11:21:52 IST
    Subject: Re:Integrals challenge 1
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elastiboysai (2327)

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Though Koni used it directly being unaware of
Euler's subst.
I'll justify the substitution.
Read the abov first
In the given problem
f(x)=7x-10-x^2
the quadratic has real roots 2 and 5
a<o
c<0
So u cant apply Euler's first and second substitution
 
Hence we use the third
Now over to the image
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[Post New]11 Mar 2008 11:21:58 IST
    Subject: Re:Integrals challenge 1
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konichiwa2x (2224)

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well radon, with sufficient practice, you'll be able to visualise "what will happen" in the next couple of steps if you make so or so substitution...
Guide to latex:
http://www.goiit.com/posts/list/community-shelf-a-guide-to-latex-48056.htm

JEE and OLYMPIA INFINATUM
http://iit-redefined.theforum.name/index.php
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[Post New]11 Mar 2008 11:37:28 IST
    Subject: Re:Integrals challenge 1
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elastiboysai (2327)

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Heres my method
alm. same as abuv
bt only I had a reason more than an instinct to substitute

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[Post New]15 Mar 2008 19:14:08 IST
    Subject: Re:Integrals challenge 1
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elastiboysai (2327)

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Hi guyz
Razz
I just found anoder interesting method 2 solv it.. widout Euler's substitution.



P = \int \frac{x}{(7x - 10 - x^2)^{3/2}}\,dx = \int \frac{x}{\left[\frac{9}{4} - \left(x - \frac{7}{2}\right)^2\right]^{3/2}}\,dx,

and now we are to make the substitution x - \frac{7}{2} = \frac{3}{2} \sin t: the result is

P = \int \frac{\frac{3}{2} \sin t + \frac{7}{2}}{\left(\frac{9}{4} - \frac{9}{4} \sin^2t\right)} \cdot \frac{3}{2} \cos t\,dt = \frac{2}{9}\int (3 \cos^{-2}t \sin t + 7 \sec^2t)\,dt

and you can finish from here.Very Happy
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