Question

calculus help

Answer 1

find \[d/dx \int\limits_{2}^{x^3}\ln(x^2)dx\]

Answer 2

thats what i have so far but idk whts next

Answer 3

@mrm

Answer 4

Since you're taking the derivative of an integral, there is a shortcut here. A derivative will undo an integral, you just need to know what the limits of integration are an plug them into the expression inside the integral.

Answer 5

Remember to do the chain rule.

Answer 6

I forgot what the chain rule is.. I'll try to do a step

Answer 7

ln(x^3)^2

Answer 8

\[d/dx \int\limits_{2}^{x^3}\ln(x^2)dx = f(x)\] we want to find f(x)

let \[G(x) = \int\limits_{2}^{x}\ln(x^2)dx\]
then
\[G(x^3) = \int\limits_{2}^{x^3}\ln(x^2)dx \]

then we can say
\[\frac{d}{dx} G(x^3) = f(x)\]

Answer 9

Then do i plug in 2 into the the equation?

Answer 10

as formulated, the answer is zero

https://i.gyazo.com/bad010ce977ce9953dba65406af53d16.png

Answer 11

I put it into my calc and got 18.5 as the answer, but I haveto show my work

Answer 12

You will not get a number. You will get some function of x... f(x).

Answer 13

You see that one of your limits of integration is a function of x, \[x^3\]

Answer 14

thanks Prof!! we just need to switch the letters about, if i can be so bold as to make a recommendation

Answer 15

when you differential the integral, you get
\[\frac{d}{dx}G(x^3) = ln(x^6)dx\]

Answer 16

right. so now we plug in 2?

Answer 17

then by the chain rule we have to take the derivative inside G... what is the derivative of \[x^3\]

Answer 18

3x^2

Answer 19

so our answer is \[f(x) = 3x^{2}\ln(x^6)\]

Answer 20

and that's the simplest form?

Answer 21

Yep.

Answer 22

Thanks for your help!

Answer 23

No problem.

Answer 24

that's just nonsense

it's actually a meaningless integral

\(\dfrac{d}{dx} \int ANYTHING ~ dx = ANYTHING\) [ftc]

the limits don't matter, and they mean nothing. so the answer is \(\ln (x^2)\)


i was wrong to be so flippant, it just looked silly, but this is wrong too.....IMHO!!!

Answer 25

Irish has a point. It should be something like \[\Large d/dx \int\limits\limits_{2}^{x^3}\ln(t^2)dt\]not the way it was originally posted.

Answer 26

yep!

no-one cares though!!