# Thread: derivative of an integral?

1. ## derivative of an integral?

$\displaystyle \int_{\sqrt{x}}^ x \frac{cos z}{z} dz$

I have no idea what to do.

2. You could try integration by parts in the form

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

or use a series approximation for $\displaystyle \cos(z)$

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

I have no idea what to do.
the Fundamental Theorem of Calculus ...

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

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

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

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

$\displaystyle \frac{d}{dx} \left[ \int_v^u f(z) \, dz \right] = f(u) \cdot \frac{du}{dx} - f(v) \cdot \frac{dv}{dx}$
I never seen a problem like this before and now thanks to you i know how to solve it when i see it again