Question

I've actually come across a question in my review packet I don't understand how to do, it's show that the function F(x) =
∫x4x1/tdt
is constant on the interval 0 to positive infinity.

Answer 1

\[\int\limits_{x}^{4x} 1/t\]

Answer 2

I think it wants you to use the fundamental theorem of calculus to find F ' (x) and show that F ' (x) is 0

When F ' (x) = 0, the function F(x) is constant

Answer 3

I realized that after I posted it, completely forgot about the fundamental theorem, it comes out to ln (4)

Answer 4

I meant this part of the FTC

If \[\Large g(x) = \int_{m(x)}^{n(x)}f(t)dt\]

then

\[\Large g \ '(x) = f \ '(n(x))*n \ '(x) - f \ '(m(x))*m \ '(x)\]

I'm using the chain rule as well

Answer 5

I think I understand that problem, do you think you could help me with another one?

Answer 6

sure

Answer 7

Find \[\int\limits_{2}^{x^3} \ln(x^2) dx\]

Answer 8

is that a typo? you used x in two different spots

Answer 9

That's how my packet has it
i'm just as confused as you are, it should be t or something right?

Answer 10

yeah it should be ln(t^2)dt

Answer 11

ok so assuming that, how would I solve this?

Answer 12

does it want you to use the FTC again?

Answer 13

oh wait, nvm I see how to do this

Answer 14

ln(t^2) = 2*ln(t) using one of the log rules

Answer 15

you can then use integration by parts to integrate ln(t)

Answer 16

This is under the fundamental theorem of calculus section of my packet, how would I use that to solve this?

Answer 17

what are the full instructions? if possible, take a screenshot or take a pic of the full problem.

also, have you learned about integration by parts yet?

Answer 18

I have learned integration of parts, It only says find (equation) but is under the section of fundamental theorem of calculus, so I assume that I need to use the theorem to solve it,