Understanding Integration with a Constant in the Limits

[vorcil]

I need to figure out,

\int_0^h \frac{1}{2\sqrt{hx}}dx

If h is a constant,

how do i do this?

my book shows that I can pull out,

\frac{1}{2\sqrt{h}} \int \frac{1}{\sqrt{x}}dx

How does the 2 from \frac{1}{2\sqrt{hx}} come out with the \sqrt{h}?

I thought I would've only been able to pull out 1/root h,

like this,

\frac{1}{\sqrt{h}} \int \frac{1}{2\sqrt{x}}dx

-

why does 2 root h get assigned constant? instead of only h

 

[rock.freak667]

1/2 is a constant, 1/√h is a constant

it must follow that

1/2√h is constant as well.

 

[Mark44]

I need to figure out,

\int_0^h \frac{1}{2\sqrt{hx}}dx

If h is a constant,

how do i do this?

my book shows that I can pull out,

\frac{1}{2\sqrt{h}} \int \frac{1}{\sqrt{x}}dx

How does the 2 from \frac{1}{2\sqrt{hx}} come out with the \sqrt{h}?

I thought I would've only been able to pull out 1/root h,

like this,

\frac{1}{\sqrt{h}} \int \frac{1}{2\sqrt{x}}dx

-

why does 2 root h get assigned constant? instead of only h

The basic idea is that \int k*f(x) dx = k*\int f(x) dx.

The rest in your problem is just algebra.
\frac{1}{2\sqrt{hx}} = \frac{1}{2*\sqrt{h}\sqrt{x}} = \frac{1}{2\sqrt{h}} \frac{1}{\sqrt{x}}

Integration is being done with respect to x (i.e., with x as the variable), so h is just another constant in this process.

 

[vorcil]

cheers