Expert: Paul Klarreich - 8/12/2009

Question
Hi,
I'm having trouble with this integral, because I'm not sure what do do since the upper limit is a variable. Here it is:

Find int( sinx/(sqrt(cosx))dx)
The limits are from 0 to x/4.

Thanks!

Answer
Questioner: Amy
Country: United States
Category: Calculus
Private: No
Subject: Calculus- Definite Integral
Question: Hi,
I'm having trouble with this integral, because I'm not sure what do do since the upper limit is a variable. Here it is:

Find int( sinx/(sqrt(cosx))dx)
The limits are from 0 to x/4.

Thanks!
----------------------------------------
Hi, Amy,

Your integral is:

{x/4   sin t
|    ---------- dt
}0   sqrt(cos t)

Yes, I know -- I seem to have changed an 'x' to a 't'.  No matter -- the following are all exactly the same:

{x/4   sin z
|    ---------- dz
}0   sqrt(cos z)

{x/4   sin s
|    ---------- ds
}0   sqrt(cos s)

{x/4   sin y
|    ---------- dy
}0   sqrt(cos y)

AND EVEN
{x/4   sin x
|    ---------- dx   << the last example.
}0   sqrt(cos x)

Get the idea?  When you write:

{something else
|                  f(a variable) d(that variable)
}something

That variable is not really a variable -- it is called a 'dummy'.  That means -- when you are all done, the dummy disappears.  Even if the dummy is 'x', as in the last example.  That x and the one in the x/4 are not the same -- they have absolutely nothing to do with one another.

So what you are looking at:

{x/4   sin t
|    ---------- dt
}0   sqrt(cos t)

is a function of x, not of t.  

Now that you know that, the rest should be easy:

A. Use a standard method (in this case let  u = cos t) to find your 'antiderivative.'

B. Substitute  t = x/4  and  t = 0,  then subtract.

You WILL get some x's in your answer.  Of course you will; you are substituting an x/4.