`F(x) = int_pi^ln(x) cos e^t dt` Find the derivative.

`F(x)=int_pi^(lnx) cos(e^t)dt`

`F'(x)=?`

Take note that if the function has a form

`F(x)=int_a^(u(x)) f(t)dt`

its derivative is

`F'(x)=f(u(x))*u'(x)`

Applying this formula, the derivative of the function

`F(x)=int_pi^(ln(x)) cos(e^t)dt`

will be:

`F'(x) = cos(e^(ln(x))) *(ln(x))'`

`F'(x) = cos(e^(ln(x)))*1/x`

`F'(x)= cos(x)*1/x`

`F'(x)= cos(x)/x`

Therefore, the derivative of the given function is `F'(x)=cos(x)/x` .