Expert: Paul Klarreich - 1/31/2010

Question
Evaluate the following integral:
{infinity
| 1/((sqrt(x))(x+1))dx
}0

Since I cannot evaluate the integral with these bounds, I wanted to make a substitution for both infinity and 0.
I changed the equation to-
lim a----> infinity integral of
{a
| 1/((sqrt(x))(x+1))dx
}1

that added to

lim b----> 0 from the right integral of
{1
| 1/((sqrt(x))(x+1))dx
}b

After this I am stuck. I don't know if this is the correct step either.

Answer
Questioner: Jamie
Country: United States
Category: Calculus
Private: No
Subject: Integrals involving limits
Question: Evaluate the following integral:
{infinity
| 1/((sqrt(x))(x+1))dx
}0

Since I cannot evaluate the integral with these bounds, I wanted to make a substitution for both infinity and 0.
I changed the equation to-
lim a----> infinity integral of
{a
| 1/((sqrt(x))(x+1))dx
}1

that added to

lim b----> 0 from the right integral of
{1
| 1/((sqrt(x))(x+1))dx
}b

After this I am stuck. I don't know if this is the correct step either
..............................................
I think the first thing is to try integrating it.  You get:


{inf
| 1/((sqrt(x))(x+1))dx = arctan(sqrt(x))
}0

Now try arctan(sqrt(x)), from 0 to infinity.

arctan(sqrt(0)) = arctan(0) = 0, no problemo.

And  lim[x-> inf] arctan(sqrt(x)) = pi/2  [just look at the graph.]

So there is your answer.