Use the second fundamental theorem of calculus to evaluate: d/(dx) int_4^(e^x) ln(t^2)e^tdt

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`d/(dx) int_4^(e^x) ln(t^2)e^tdt`


The second fundamental theory of calculus tells us,

`d/(dx) int_a^x f(t)dt = f(x)`

here we need to do a substitution, as , `u = e^x`

then it gives, `(du)/(dx) = e^x = u`

let,  `y = int_4^(e^x) ln(t^2)e^tdt`

then we have to find `(dy)/(dx)`

we will convert this with our substitution,

`y = int_4^u ln(t^2)e^tdt`

now, from the second theorem,

`(dy)/(du) = f(u) = ln(u^2)e^u`

so we apply chain rule to find `(dy)/(dx)`

`(dy)/(dx) = (dy)/(du) * (du)/(dx)`

`(dy)/(dx) = ln(u^2)e^u*e^x`


`(dy)/(dx) = ln(e^(2x))e^(e^x)*e^x`

`(dy)/(dx) = 2x*e^(e^x)*e^x`



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