Analyzing the Definite Integral (1+x)^(-1/2)

[SW VandeCarr]

Is there an analytic solution to the following integral? (Not a homework question, solved numerically).

$$\int_{0}^{\infty}{\frac{dx}{(1+x)\sqrt{x}}=\pi$$

 

[l'Hôpital]

Let u^2 = x

 

[SW VandeCarr]

Let u^2 = x

OK I get: $$2\int\frac{1}{1+u^2}du$$

I'm a bit rusty at this. I think I need to make some trig substitutions here, but I'm at a loss as to exactly what.

 

[jrlaguna]

It's just there, the arc tangent.

 

[SW VandeCarr]

It's just there, the arc tangent.

Right. I finally saw that. However, the solution: $$2\arctan(\sqrt{x})+C$$ contains a variable. The solution given in the CRC Standard Mathematical Tables 14th ed, page 342, Eq 489 is $$\pi$$.

EDIT:Yes, I know it's old, but I doubt a new solution has been invented. I bought it for $2.

 

[D H]

You looked up the indefinite integral. You have a definite integral. Apply the integration limits.

 

[SW VandeCarr]

You looked up the indefinite integral. You have a definite integral. Apply the integration limits.

The form of the integral in the book is just as I wrote it in the first post, but I see how the solution $$\pi$$ is obtained. Thanks.