Indefinite Integration of e^t ()

[shrill_kill]

Can somebody explain how e^t(1/(t^3) - 3/(t^4) ) dt = e^t/(t^3) + c
(please use an image if possible)​

 

[Deleted member 4993]

View attachment 33635


Can somebody explain how e^t(1/(t^3) - 3/(t^4) ) dt = e^t/(t^3) + c
(please use an image if possible)​

Can you integrate:

\(\displaystyle \int{\frac{3e^t}{t^4}}dt\)

 

[shrill_kill]

Can you integrate:

\(\displaystyle \int{\frac{3e^t}{t^4}}dt\)

No

 

[Steven G]

You have to try. Make some substitutions, try integration by parts--something.
No helper on this forum will solve your problem for you as that would not be helpful. Helper will help you get to the correct solution but 1st need to know what you have tried and where you are stuck.

 

[Deleted member 4993]

No

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem

 

[topsquark]

Can you integrate:

\(\displaystyle \int{\frac{3e^t}{t^4}}dt\)

No

Hint: Integrate [imath]\displaystyle \int \dfrac{-3 e^t}{t^4} ~ dt[/imath] by parts once. Do you see anything in the whole problem that might cancel?

-Dan

Addendum: We have to be able to understand what you are posting! The second line in your OP needs commas between the statements.

 

[Dr.Peterson]

View attachment 33635


Can somebody explain how \(e^t(1/(t^3) - 3/(t^4) ) dt = e^t/(t^3) + c\)
(please use an image if possible)​

It would be more useful for you to use an image (of handwritten or properly typed work!), than for us to do so, since your image is almost unreadable. Trying to interpret what you wrote here, I think it's this:

[math]\int\frac{\log x-3}{(\log x)^4}dx\\=\int\frac{t-3}{t^4}e^tdt\\=\int e^t\left(\frac{1}{t^3}-\frac{3}{t^4}\right)dt\\=\frac{e^t}{t^3}+c\\=\frac{x}{(\log x)^3}+c[/math]
You are asking about the middle step. Are you implying that this is a solution you were given that you don't understand? There is indeed a leap there, which I would expect to have been explained at least with a hint.

The natural thing to do is to integrate [imath]\displaystyle\int \frac{e^t}{t^3}dt[/imath] and [imath]\displaystyle\int \frac{-3e^t}{t^4}dt[/imath], each by parts. But do the latter first, and you'll see that things work out very nicely. I assume you know how to use parts.

 

[Steven G]

View attachment 33635
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Which is it, dt=dx or dx=xdt or are they both correct because x=1?

 

[Dr.Peterson]

Which is it, dt=dx or dx=xdt or are they both correct because x=1?

To clarify this, it's a reference to

​

which is nonsense. You probably meant something like this:

​

The equal sign has a specific meaning, and must not be used where you mean something else.