Evaluate the following derivatives.

$\frac{d}{dx}{\int}_{3}^{{e}^{x}}{\mathrm{cos}t}^{2}dt$

tinfoQ
2021-02-13
Answered

Evaluate the following derivatives.

$\frac{d}{dx}{\int}_{3}^{{e}^{x}}{\mathrm{cos}t}^{2}dt$

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Talisha

Answered 2021-02-15
Author has **93** answers

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,

$\int}_{3}^{{e}^{x}}{\mathrm{cos}t}^{2}dt={\mathrm{cos}\left({e}^{x}\right)}^{2}\frac{d\left({e}^{x}\right)}{dx}-{\mathrm{cos}\left(3\right)}^{2}\frac{d\left(3\right)}{dx$

$\Rightarrow {\int}_{3}^{{e}^{x}}{\mathrm{cos}t}^{2}dt={\mathrm{cos}e}^{2x}\left({e}^{x}\right)-\mathrm{cos}9\left(0\right)$

$\therefore {\int}_{3}^{{e}^{x}}{\mathrm{cos}t}^{2}dt={e}^{x}\mathrm{cos}\left({e}^{2x}\right)$

To evaluate the derivative,

Step 2

According to the Lebnitz Rule of differentiation under the sign of integration,

Step 3

Let us apply the above rule,

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