# Thread: [SOLVED] derivative of integral

1. ## [SOLVED] derivative of integral

Hello, I want to calculate the derivative of f(x), where:

$\displaystyle f(x)=\int^x_{sin{x}} e^{t^2}dt$

I don't have many ideas how to start with this, I had thought maybe transforming it into a taylor series and then evaluate and calculate its derivative, what do you think?

Thank you.

2. Originally Posted by Bop
Hello, I want to calculate the derivative of f(x), where:

$\displaystyle f(x)=\int^x_{sin{x}} e^{t^2}dt$

I don't have many ideas how to start with this, I had thought maybe transforming it into a taylor series and then evaluate and calculate its derivative, what do you think?

Thank you.

Let $\displaystyle F(x)$ be any primitive function of $\displaystyle e^{t^2}$ (why does it exist?) , then $\displaystyle f(x)=\int\limits^x_{sin{x}} e^{t^2}dt=F(x)-F(\sin x)\Longrightarrow f'(x)=F'(x)-\cos xF'(\sin x)=$ ...

The above is just a simple particular case of the Leibnitz (differentiation under the sign of) integral rule.

Tonio

3. Ok, thank you very much.