Question

Find and simplify the derivative of the function
lnx
f(x)=∫ 〖t/e^t dt〗
1

Answer 1

fundamental theorem of calculus

Answer 2

\[\frac d{dx}F(x)=\frac d{dx}\left(\int_{g(x)}^{h(x)}f(t)dt\right)= F'(h(x))h'(x)- F'(g(x))g'(x)\]where\[F'(x)=f(x)\]

Answer 3

not that if either bound is a constant you get, for instance\[\frac d{dx}\left(\int_a^xf(t)dt\right)=F'(x)-\cancel{F'(a)}^{\large0}=F'(x)=f(x)\]

Answer 4

note*

Answer 5

And just to add a quick generalization in case you ever get a case like this again:
\[\frac{d}{dx}\int_{u(x)}^{v(x)}f(t)dt=\frac{d}{dx}(F(v(x))-F(u(x)))=f(v(x))v'(x)-f(u(x))u'(x)\]You'll also see this is consistent with what @TuringTest did. If u(x) is a constant, then u'(x) becomes zero and that whole term is crossed off.