Question

derivative of integral

Answer 1

\[\frac{ dy }{ dx }\int\limits_{1/x}^{\pi} \cos ^{3}t\]

Answer 2

Ok let's do something in general
\[\text{ Let } g(x)=\int\limits_{a(x)}^{b(x)}f(t)dt\]

Say the antiderivative of f is F then

\[g(x)=F(t)|_{a(x)}^{b(x)}\]
so
\[g(x)=F(b(x))-F(a(x))\]

Your objective is to find the derivative of g with respect to x though...
all we need to know is chain rule...
\[g'(x)=b'(x) \cdot F'(b(x))-a'(x) \cdot F'(a(x))\]
Then
Recall F'=f
\[g'(x)=b'(x) \cdot f(b(x))-a'(x) \cdot f(a(x))\]

Answer 3

sweet, thanks!

Answer 4

You're good with that?
You understand with no questions?

Answer 5

kinda. basically you want to plug the limits of integration into the function, and then take the derivative using chain rule?

Answer 6

but also in the answer we got we didn't even need to find the antiderivative because nowhere in that formula did we have F
we had F' which was f

Answer 7

Like you can let f(t)=cos^3(t)
let a(x)=1/x
let b(x)=pi

Answer 8

find a' and b'

Answer 9

and then plug away

Answer 10

You could use trig identity for your problem
\[\cos^2(t) = \frac{1+\cos(2t)}{2}\]

Answer 11

I thinks it cube not squared
also he doesn't need to do any integration for his problem