# Evaluate the following derivatives. \frac{d}{dx}\int_{3}^{e^{x}}\cos t^{2}dt

Evaluate the following derivatives.
$\frac{d}{dx}{\int }_{3}^{{e}^{x}}{\mathrm{cos}t}^{2}dt$
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Step 1
To evaluate the derivative,
$\frac{d}{dx}{\int }_{3}^{{e}^{x}}{\mathrm{cos}t}^{2}dt$
Step 2
According to the Lebnitz Rule of differentiation under the sign of integration,
$\frac{d}{dx}{\int }_{g\left(x\right)}^{h\left(x\right)}f\left(t\right)dt=f\left(h\left(x\right)\right)\frac{dh\left(x\right)}{dx}-f\left(g\left(x\right)\right)\frac{dg\left(x\right)}{dx}$
Step 3
Let us apply the above rule,
${\int }_{3}^{{e}^{x}}{\mathrm{cos}t}^{2}dt={\mathrm{cos}\left({e}^{x}\right)}^{2}\frac{d\left({e}^{x}\right)}{dx}-{\mathrm{cos}\left(3\right)}^{2}\frac{d\left(3\right)}{dx}$
$⇒{\int }_{3}^{{e}^{x}}{\mathrm{cos}t}^{2}dt={\mathrm{cos}e}^{2x}\left({e}^{x}\right)-\mathrm{cos}9\left(0\right)$
$\therefore {\int }_{3}^{{e}^{x}}{\mathrm{cos}t}^{2}dt={e}^{x}\mathrm{cos}\left({e}^{2x}\right)$