How do I find dF(x)/dx where
F(x) = integral sin(t^2) dt
[from 2^x to 3+x^3]
I'm not sure what dF(x) is (like the notation).
Just in case a picture helps...
dF(x)/dx means the derivative, with respect to x, of the function F(x), i.e. d/dx of F(x).
You were also thrown by the composition notation in Plato's formula. Below, I shall bother you with another notation for composition of functions, which you won't necessarily find any easier. However, do note that Plato's formula relies on the fact that...
$\displaystyle \int_{x}^{y}\ f(t)\ dt\ =\ \int_a^{y}\ f(t)\ dt\ -\ \int_a^{x}\ f(t)\ dt$
... for some constant a. So you may as well split your problem into two...
$\displaystyle \int_{2^x}^{3+x^3}\ sin(t^2)\ dt\ =\ \int_a^{3+x^3}\ sin(t^2)\ dt\ -\ \int_a^{2^x}\ sin(t^2)\ dt$
Now you apply Fundamental theorem of calculus - Wikipedia, the free encyclopedia, which kind of says...
... if the straight line means differentiating (downwards) with respect to x (the continuous line) or with respect to the dashed balloon (the dashed line). The really confusing bit here, with or without the diagram, is the 'dummy variable', t. I googled a helpful explanation of this, can't find it right now. But the left diagram does basically tell it like it is, i.e. when differentiating the integral with respect to the upper limit (x) you get just the integrand said about x instead of about t or whatever.
The dashed is because we might need...
... the chain rule, as we do here. Writing...
$\displaystyle \int_a^{3+x^3}\ sin(t^2)\ dt$
... as...
... and then working downwards through the chain rule, we get...
And you might like to see if you can do similarly for the second part - i.e. the blanks here...
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