studos
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integral of xe^(2lnx)
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TeeEm
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(Original post by studos)
integral of xe^(2lnx)
that is x3 to be integrated
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appliedmathlete
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0.25x^4 + c?
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NotLandau
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e^(2lnx) is simply x^2, so to integrate that function will simply boil down to determining the integral of x^3, which is obvious.
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studos
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how exactly
e^(2lnx) is simply x^2 ?
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Zacken
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(Original post by studos)
how exactly
e^(2lnx) is simply x^2 ?
2\ln x = \ln x^2, e and ln are inverse functions.

You know for example that \sqrt{f(x)^2} = f(x) (domain blah blah, shhh) because sqrt(x) and x^2 are inverse functions. So when you apply one to the other you get the argument.

In this case: \displaystyle e^{\ln x^2} = x^2. In general: \displaystlye e^{\ln f(x)} = f(x)
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studos
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ok but can I solve the integral with a method without knowing that e^lnx=lnx?

also, I need proof of e^lnx=lnx

if e^lnx=a, then lna=lnx, so not a=lnx
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Zacken
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(Original post by studos)
ok but can I solve the integral with a method without knowing that e^lnx=lnx?

also, I need proof of e^lnx=lnx

if e^lnx=a, then lna=lnx, so not a=lnx
Who said e^ln x = ln x, we're all saying e^ln x = x
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WhiteGroupMaths
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The proof is rather straightforward.

Let y= e^ (lnx)

Taking the natural logarithm on both sides gives

ln y= ln [ e^ (lnx) ] = (ln x) * lne =ln x since lne=1

As such, since ln y=lnx, then y=x= e^ (lnx) (shown)

Hope this helps. Peace.
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