# Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. int_(e^x)^0 2sin^(2)t dt ?

Jun 24, 2018

From FTC part 1:

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

And applying chain rule:

$\frac{d}{\mathrm{dx}} {\int}_{g \left(x\right)}^{a} f \left(t\right) \mathrm{dt} = - f \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

So:

$\frac{d}{\mathrm{dx}} {\int}_{{e}^{x}}^{0} 2 {\sin}^{2} t \setminus \mathrm{dt}$

$= - 2 {\sin}^{2} \left({e}^{x}\right) \frac{d \left({e}^{x}\right)}{\mathrm{dx}}$

$= - 2 {e}^{x} {\sin}^{2} \left({e}^{x}\right)$