# How do you use the Fundamental Theorem of Calculus to find the derivative of f(x)=int_1^(x)sqrt(e^t+sin(t))dt ?

Oct 9, 2014

Fundamental Theorem of Calculus

$\frac{d}{\mathrm{dx}} {\int}_{a}^{x} f \left(t\right) \mathrm{dt} = f \left(x\right)$

This theorem illustrates that differentiation can undo what has been done to $f$ by integration.

Let us now look at the posted question.

$f ' \left(x\right) = \frac{d}{\mathrm{dx}} \setminus {\int}_{1}^{x} \sqrt{{e}^{t} + \sin t} \mathrm{dt} = \sqrt{{e}^{x} + \sin x}$

I hope that this was helpful.