Question

calc help, integral [e^4 , e] 1/xlnx^(3)

Answer 1

\(\int\limits_{e}^{e^4}\frac{dx}{x \ln x^3}\) or \(\int\limits_{e}^{e^4}\frac{dx}{x \ln^3 x}\)

Answer 2

first one

Answer 3

id use u substitution for that right?

Answer 4

Yes. Can you guess what substitution you need?

Answer 5

lnx

Answer 6

dx=dux

Answer 7

You are right. Don't forget to change limits while making substitution.

Answer 8

what do you mean by that?

Answer 9

If I have allowance to ask something, I'd try to use
\[u= ln(x^3)\] as a substitution, is that what you meant kingpin?

Answer 10

I had left the ^3 in the equation. is that also going to be substituted?

Answer 11

Because then you would obtain:

\[ \frac{du}{dx}=\frac{1}{x^3} \cdot 3x^2 \\
\text{or} \\dx=\frac{x}{3}du \]

Answer 12

then you have

\[ \frac{1}{3} \int_a^b \frac{1}{x \cdot u } \cdot \frac{x}{3} du \]

Answer 13

\[u=\ln x^3, \ du=\frac{1}{x^3}\cdot 3x^2dx=\frac{3}xdx\]

Answer 14

exactly @klimenkov

Answer 15

But there were limits for one function, and now you have another, so you will have new limits.

Answer 16

one sec just working it out so i can see it myself

Answer 17

the derivative of ln(x)^3 = 1/x^3 correct

Answer 18

I dislike being a smarts but in this case it's needed.

ln(x)^3 is not the same as ln(x^3)

Answer 19

ohhhhhhhh, it would be 3(1/x)^2

Answer 20

well yes, but in the equation it's

\[ \ln(x^3) \]
so the derivative is

\[ \frac{1}{x^3} \cdot 3x^2 \]

Answer 21

sorry if im being difficult i just learned this today

Answer 22

you are doing fine, just showing empathy for the accurate way of typing it out, because the brackets can make a huge difference (-:

Answer 23

basically you are using the chain rule then

Answer 24

exactly!

Answer 25

Example:
\[\int\limits_{1}^{e}\frac{\ln x}{x}\ dx\]Substitution: \(u=\ln x, \ du=\frac{dx}{x}\)
New limits: \(a_1=u(a)=\ln 1=0, \ b_1=u(b)=\ln e=1\)
Result:
\[\int\limits_{1}^{e}\frac{\ln x}{x}\ dx=\int\limits_{0}^{1}u\ du\]

Answer 26

alright so after finding the dx the new equation would be \[3 \int\limits_{e}^{e^4} du/xu\]

Answer 27

which is the same as \[3\int\limits_{e}^{e^4} \left| u \right| /x\]?

Answer 28

Thats not right. You have to get rid of x.

Answer 29

the x was canceled from before?

Answer 30

See my example and yours. I have no x and you have.

Answer 31

You can make the integral like in my example from yours problem by making something with logarithm. What do you have to do?

Answer 32

i just have to evaluate the equation

Answer 33

Try to transform \(\ln x^3\).

Answer 34

ok i just redid the whole problem what i cam up with is \[1/3\int\limits_{e}^{e^4} du \div u\]

Answer 35

i know that the integral of du/u is \[\left| u \right|\]

Answer 36

no, the integral of du/u is ln|u|

Answer 37

yeah i just forgot to put the ln

Answer 38

hehe ok, good then.

Answer 39

if im using the e^4 and e it would look like (1/3) ln (ln(x^3)) - (1/3)ln(lnx^3))?

Answer 40

well done, yes.

Answer 41

and i can straight plug n chug in the calculator right?

Answer 42

yes you can.