Question

Need help with calculus problem involving the second fundamental theorem of calculus

Answer 1

\[\int\limits_{1}^{x^2}(1/t) dt\]

Answer 2

F(x) = 1 and im supposed to evaluate the problem above

Answer 3

my problem is that I know for the second fundamental theorem of calc im supposed to replace t with the x^2 but then i still need to integrate so do i integrate from 1 to u?

Answer 4

\[\int\limits_{1}^{u} (1/x^2)2x dx\]

Answer 5

is that correct or do i still keep the x^2 on the top?

Answer 6

can u provide a scan of the prob? seems like something is wrong or missing from what u posted...

Answer 7

sorry i dont have a scan but the problem is exactly like that

Answer 8

that integral at the top is the same as F(x) which also equals 1

Answer 9

is it F(x)=1 or F(1)=1?

Answer 10

F(x) = 1

Answer 11

oh.....so the question is asking u to solve for x then?

Answer 12

im assuming so because it just says to solve the integral

Answer 13

hahahaa its kinda confusing....i thought it was to find the integral or something like that.

lets assume it is asking to solve for x okay?

Answer 14

ok

Answer 15

would it be a good idea to integrate first and then plug in x^2 instead of switching x^2 and t?

Answer 16

Correct @YadielG

Answer 17

do the integral then plug in x^2 and 1 to evaluate the integral.

Answer 18

let \[G(t) = \int\limits_{}^{}\frac{ 1 }{ t}dt,\]

F(x) = 1 = G(x^2) - G(1)

Then solve for x.

Answer 19

is there a good rule of thumb for when to actually use the second fundamental theorem of calc? like only use it for definite integrals where the solution is not given

Answer 20

@YadielG in this case, ASSUMING the prob is to solve for x, u dont need the second fundamental theorem as u are not trying to find F'(x).

Answer 21

That makes sense, maybe im just over thinking things lol

Answer 22

thank you for your help

Answer 23

err...the operation is to integrate...not differentiate on the 1/t part....so G(t) is not -1/t^2

Answer 24

super dave.... thank you☘

\(\large \int\limits_{1}^{x^2} dt \quad (\frac{1}{t} )\)

\(\large= \left[\ln t\right]_{1}^{x^2} = 2 \ln x\)

Answer 25

@IrishBoy123 welcome

@YadelG let us know if u need more help to solve for x. good luck :)