integrate this

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Blazing goIITian

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I will have a go.. :D

Let, $I=int rac{(x-1)dx}{(x+1)sqrt{x^3+x^2+x}}$

$=int rac{(x^2-1)dx}{(x+1)^2sqrt{x^3+x^2+x}}\\=int rac{x^2(1-1/x^2)dx}{(x^2+2x+1)sqrt{x^3+x^2+x}}$

Let,

$x+ rac{1}{x}=t\\Rightarrow (1-1/x^2)dx=dt$

Now we have,I= int rac{dt}{(t+2)sqrt{t+1}}

Let t+1=z^2 herefore dt=2z dz

so , $int rac{dt}{(t+2)sqrt{t+1}}=int rac{2z dz}{(z^2+1)sqrt{z^2}}=2int rac{dz}{z^2+1}$

$=2 an^{-1}(z)+C=2 an^{-1}(sqrt{t+1})+C=2 an^{-1}sqrt{ rac{x^2+x+1}{x}}+C$

I hope it is correct .....

[IMG]http://alt1.artofproblemsolving.com/Forum/latexrender/pictures/0/6/3/0638cc54ee34d87aa803d78fe3ec072aa8749bf5.gif[/IMG] :)

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