# derivative of an integral?

• Oct 8th 2009, 04:29 PM
yoman360
derivative of an integral?
$\int_{\sqrt{x}}^ x \frac{cos z}{z} dz$

I have no idea what to do.
• Oct 8th 2009, 04:36 PM
pickslides
You could try integration by parts in the form

$\int_{\sqrt{x}}^ x z^{-1}\cos( z) ~dz$

or use a series approximation for $\cos(z)$
• Oct 8th 2009, 04:37 PM
skeeter
Quote:

Originally Posted by yoman360
$\int_{\sqrt{x}}^ x \frac{cos z}{z} dz$

I have no idea what to do.

the Fundamental Theorem of Calculus ...

$u$ and $v$ are both functions of $x$

$\frac{d}{dx} \left[ \int_v^u f(z) \, dz \right] = f(u) \cdot \frac{du}{dx} - f(v) \cdot \frac{dv}{dx}$
• Oct 8th 2009, 04:56 PM
yoman360
Quote:

Originally Posted by skeeter
the Fundamental Theorem of Calculus ...

$u$ and $v$ are both functions of $x$

$\frac{d}{dx} \left[ \int_v^u f(z) \, dz \right] = f(u) \cdot \frac{du}{dx} - f(v) \cdot \frac{dv}{dx}$

I never seen a problem like this before and now thanks to you i know how to solve it when i see it again(Rofl)