# Math teacher trying to be funny

• May 17th 2008, 02:00 PM
rexexdesign
Math teacher trying to be funny
Hi everybody, is it too hot to go ouside where you live as well?

My teacher gave us this problem and said something about the derivative of the antiderivative is the function itself. Here is the function:

d/dx integral from x^1/3 to x^2 ( csc y / y ) dy
(sorry I don't know how to make the integral look better on my iPhone)

If anyone can give me a hint on where to start that would be great.

Thanks!
Alex
• May 17th 2008, 02:13 PM
galactus
The second fundamental theorem of calc:

$\displaystyle \frac{d}{dx}\int_{h(x)}^{g(x)}f(t)dt=f(g(x))g'(x)-f(h(x))h'(x)$

So, you have:

$\displaystyle \frac{csc(x^{2})}{x^{2}}(2x)-\frac{csc(x^{\frac{1}{3}})}{x^{\frac{1}{3}}}(\frac {1}{3}x^{\frac{-2}{3}})$

$\displaystyle =\frac{6sin(x^{\frac{1}{3}})-sin(x^{2})}{3xsin(x^{\frac{1}{3}})sin(x^{2})}$
• May 18th 2008, 06:09 PM
rexexdesign
thanks
You are the best, I filled in the simplifications and got the answer just like you, that was very helpful.

Thanks again.