Question

find the value of the integral 1_^64 ((log4 x)^2 −√3 log4 x +7)/x dx

Answer 1

put \(u=\log_4(x)\)

Answer 2

change the limit of integration while you are at it, so you don't have to change back at the end
\\[u=\log_4(x),u(1)=\log_4(1), u(64)=\log_4(64)\] both are cooked up to be easy

Answer 3

⌠64
⌡1 ((log4 x)^2 −√3 log4 x +7)/x dx


i am still abit lost @satellite

Answer 4

you know the derivative of \(\log_4(x)\)?

Answer 5

if you do not, i can tell you

Answer 6

no i dont

Answer 7

do you know the derivative of \(\ln(x)\)?

Answer 8

1/x right ?

Answer 9

yes

Answer 10

and by the all mighty "change 'o base" formula \[\log_4(x)=\frac{\ln(x)}{\ln(4)}\] so the derivative is \[\frac{1}{x\ln(4)}\]

Answer 11

keeping in mind that \(\ln(4)\) is just a constant (some number)

Answer 12

make the \(u\) sub and you have it in one step basically

Answer 13

\[\frac{1}{\ln(4)}\int_0^3(u^2-\sqrt{3}u+7)du\] in other words

Answer 14

I understand now, then from taking out that constant I will be able to anti differentiate?

Answer 15

@satellite

Answer 16

yes use the backwards power rule for each term

Answer 17

i get the feeling that the process was a bit of a mystery
want to do it real slow?

Answer 18

yeah that will be great, would make me understand even more

Answer 19

what did you do with the x the divides the term ?

Answer 20

it is a set up for a u-sub because each term has a \(\log_4(x)\) in it
plus the derivative of \(\log_4(x)\) is right there in front of you, although hiding a little bit

Answer 21

\[\int\frac{\log_4(x)^2-\sqrt3\log_4(x)+7}{x}dx\]

Answer 22

think of it like this \[\int \log_4(x)-\sqrt3\log_4(x)-7)\times \frac{dx}{x}\]

Answer 23

well actually like this \[\int( \log_4(x)^2-\sqrt3\log_4(x)-7)\times \frac{dx}{x}\]

Answer 24

\(\frac{1}{x}\) is almost the derivative of \(\log_4(x)\)
put \(u=\log_4(x)\) then \[du=\frac{dx}{x\ln(4)}\]

Answer 25

the constant in the denominator doesn't make any difference, it comes right out front
you get \[\frac{1}{\ln(4)}\int (u^2-\sqrt{3}u+7)du\]

Answer 26

i am assuming that you have seen a u-sub before, just not one as not so obvious as this one
for example if you had
\[\int \sin^2(x)\cos(x)dx\] y ou would make \[u=\sin(x),du=\cos(x)dx\] and then integrate \[\int u^2du\]

Answer 27

understand that part now

Answer 28

then while you are at it, change the limits of integration so you don't have to revert back to \(\log_4(x)\) at the end
you got that part?

Answer 29

what do you mean change the limits ?

Answer 30

@satellite

Answer 31

thanks i clearly understand it and got the ans @satellite