Use the second fundamental theorem of calculus to evaluate:

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

Expert Answers

An illustration of the letter 'A' in a speech bubbles

`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`

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial