Problem with the answer provided by wolfram alpha

**flintstone** · January 6th 2010, 12:43 AM

I am having a problem with evaluating the integral so i put the question in it and i got a answer but i am not sure of his steps

$\int \frac{1}{x(1-x^{\frac{1}{4}})}$

http://www.wolframalpha.com/input/?i=integrate+dx%2F+(x(1-x^(1%2F4)))

I didnt understand how he substituted 'u' in the equation.

**TwistedOne151** · January 6th 2010, 01:54 AM

Have you considered the sustitution $u=x^{1/4}$ ?

-Kevin C.

**Prove It** · January 6th 2010, 01:58 AM

$\int{\frac{1}{x(1 - x^{\frac{1}{4}})}\,dx}$

Let $u = x^{\frac{1}{4}}$ .

Then $\frac{du}{dx} = \frac{1}{4}x^{-\frac{3}{4}} = \frac{1}{4x^{\frac{3}{4}}}$ .

So $dx = 4x^{\frac{3}{4}}\,du$ .

Can you see that $\frac{dx}{x} = \frac{4x^{\frac{3}{4}}\,du}{x}$

$= 4x^{-\frac{1}{4}}\,du$

$= \frac{4\,du}{x^{\frac{1}{4}}}$

$= \frac{4\,du}{u}$ .

So the integral becomes

$4\int{\frac{1}{u(1 - u)}\,du}$ .

Can you go from here?

**tom@ballooncalculus** · January 6th 2010, 02:21 AM

Same thing dressed (only very) slightly different...

$\frac{1}{(1 - (x^{\frac{1}{4}}))\ x}$

$=\frac{1}{(1 - (x^{\frac{1}{4}}))\ (x^{\frac{1}{4}})^4}$

$=\frac{1}{(1 - (x^{\frac{1}{4}}))\ (x^{\frac{1}{4}})^4}\ \ \frac{\frac{1}{4}(x^{\frac{1}{4}})^{-3}}{\frac{1}{4}(x^{\frac{1}{4}})^{-3}}$

$=\frac{1}{(1 - (x^{\frac{1}{4}}))\ (x^{\frac{1}{4}})^4\ \frac{1}{4}(x^{\frac{1}{4}})^{-3}}\ \ \frac{1}{4}(x^{\frac{1}{4}})^{-3}$

$=\frac{4}{(1 - (x^{\frac{1}{4}}))\ (x^{\frac{1}{4}})}\ \ \frac{1}{4}(x^{\frac{1}{4}})^{-3}$

$=\frac{4}{(1 - (x^{\frac{1}{4}}))\ (x^{\frac{1}{4}})}\ \ d(x^{\frac{1}{4}})$

Edit:

Then partial fractions, or else this trick...

$=4\ [\ \frac{1 - (x^{\frac{1}{4}}) + (x^{\frac{1}{4}})}{(1 - (x^{\frac{1}{4}}))\ (x^{\frac{1}{4}})}\ ]\ \ d(x^{\frac{1}{4}})$

$=4\ [\ \frac{(1 - (x^{\frac{1}{4}}))}{(1 - (x^{\frac{1}{4}}))\ x^{\frac{1}{4}}} + \frac{x^{\frac{1}{4}}}{(1 - (x^{\frac{1}{4}}))\ x^{\frac{1}{4}}}\ ]\ \ d(x^{\frac{1}{4}})$

$=4\ [\ \frac{1}{(x^{\frac{1}{4}})} + \frac{1}{1 - (x^{\frac{1}{4}})}\ ]\ \ d(x^{\frac{1}{4}})$

Also, just in case an overview helps...

... where

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

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