Question

Calculus help plz!

Answer 1

@mrm

Answer 2

alright, let me take a look at it

Answer 3

@AloneS

Answer 4

do you know that
\[\int\limits g'(x) dx= g(x)+C \\ \]
and also that
\[\int\limits_a^b g'(x) dx=g(x)|_a^b =g(b)-g(a)\]

Answer 5

how do i apply this to my problem?

Answer 6

\[\text{ Let } F(x)=\int\limits_1^x g'(t) dt \\ \text{ Then we can write } F(x)=g(t)|_1^x \\ \text{ which means } F(x)=g(x)-g(1)\]

Answer 7

now to find the derivative of F
just differentiate both sides

Answer 8

\[F(x)=\int\limits_1^x g'(t) dt \implies F'(x)=g'(x)\]

Answer 9

this is one of the fundamental theorems of calculus

Answer 10

im still a little confused. can you keep helping me?

Answer 11

may i ask what part what I said is confusing so i can explain that part more

Answer 12

what am i taking the derivative of?

Answer 13

you are taking the derivative of F
aka the derivative of the integral

Answer 14

with the limits?

Answer 15

yes your integral has limits

Answer 16

the lower limit is x^3
while the upper limit is 1

Answer 17

can you tell me if you understand everything I said above?

Answer 18

yes, but i cant take the derivative of it with t

Answer 19

\[F(x)=\int\limits_1^x g'(t) dt= g(t)|_1^x =g(x)-g(1)\]
so you understood this?

Answer 20

no

Answer 21

Sorry, I lost your question Help. You're in good hands with myininaya.

Answer 22

do you know to find a definite integral you must find an antiderivative of the integrand

an antiderivative of g'(t) is g(t)

Answer 23

however what I'm trying to show is you don't actually need to know the antiderivative to find the derivative of a definite integral that is in terms of x

Answer 24

like here is another example:

\[F(x)=\int\limits_{x^2}^{1} g'(t) dt= g(t)|_{x^2}^{1}=g(1)-g(x^2) \\ \text{ now \to find } F' \\ \text{ I differentiate both sides of } F(x)=g(1)-g(x^2)\]

Answer 25

\[F'(x)=0-(x^2)' \cdot g'(x^2) \\ F'(x)=-2x g'(x^2)\]

Answer 26

g'(x^2) can be evaluated easily since you know what g'(t) equals
this was the integrand

Answer 27

can you try yours
forgot about cos(t^4) for a sec
and replace it with g'(t)

Answer 28

and follow what I did above except with your limits

Answer 29

i will try