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From: Franz Neumann on 12 Jun 2006 12:22
Hello,

for some exercises I tried to integrate the following function step by step:

1 / (sqrt(x) * (x+1)

intvx then gives me:

1) sqrt(x) / (x + x^2)
2) Square root
sqrt(x)
3) Rational fraction
0
4) 2x^2 * 1 / ((x^2+(x^2)^2)

and the solution:
2*ATAN(sqrt(x))

Now my question is what my hp is doing in step (2): Square root sqrt(x) ??

Thanks!
Franz
From: Fabian on 14 Jun 2006 08:12


On Mon, 12 Jun 2006, Franz Neumann wrote:

> Hello,
>
> for some exercises I tried to integrate the following function step by step:
>
> 1 / (sqrt(x) * (x+1)
>
> intvx then gives me:
>
> 1) sqrt(x) / (x + x^2)
> 2) Square root
> sqrt(x)
> 3) Rational fraction
> 0
> 4) 2x^2 * 1 / ((x^2+(x^2)^2)
>
> and the solution:
> 2*ATAN(sqrt(x))
>
> Now my question is what my hp is doing in step (2): Square root sqrt(x) ??
>

It is doing a substitution u=sqrt(x).

However, it is better to perform the substitution right
away, as du=1/(2*sqrt(x)):

\int 1/(sqrt(x)*(x+1)) dx = \int 2/(u^2+1) du = 2 \arctan(u)

Cheers,
Fabian
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