Final Formula: What is it and how can I find it for a given integral?

[mathemagician]

What does it mean when a question asks you to get a final formula?

I have an example of the question that is asking for this.

1. a) Find a reduction formula for $$\int(\ln x)^n dx$$ which I did find and got the answer: $$x (\ln x)^n - n \int (\ln x)^{n-1} dx$$.

But question part b) asks if I can use what I have in a) to get a final formula for $$\int(\ln x)^n dx$$ ? I do not understand what do they mean by a final formula?

Thanks for any help.

 

[phoenixthoth]

The formula is a final formula if it has NO INTEGRALS in it. So it will just have an algebraic combination of lnx and x.

 

[dextercioby]

What does it mean when a question asks you to get a final formula?
I have an example of the question that is asking for this.
1. a) Find a reduction formula for $$\int(\ln x)^n dx$$ which I did find and got the answer: $$x (\ln x)^n - n \int (\ln x)^{n-1} dx$$.
But question part b) asks if I can use what I have in a) to get a final formula for $$\int(\ln x)^n dx$$ ? I do not understand what do they mean by a final formula?
Thanks for any help.

You're asked for a recurrence formula,which u already found.Denoting
$$I_{n}=:\int (\ln x)^{n} dx$$
,u have forund that:
$$I_{n}=x(\ln x)^{n}-nI_{n-1}$$
.Make $n\rightarrow n-1$,and get:
$$I_{n-1}=x(\ln x)^{n-1}-(n-1)I_{n-2}$$
and so on,until
$$I_{0}=x(\ln x)^{0}-0I_{-1}=x$$
Use te reccurence relations to find $I_{n}$ as a function of "x" and "n".

Daniel.

 

[mathemagician]

You're asked for a recurrence formula,which u already found.Denoting
$$I_{n}=:\int (\ln x)^{n} dx$$
,u have forund that:
$$I_{n}=x(\ln x)^{n}-nI_{n-1}$$
.Make $n\rightarrow n-1$,and get:
$$I_{n-1}=x(\ln x)^{n-1}-(n-1)I_{n-2}$$
and so on,until
$$I_{0}=x(\ln x)^{0}-0I_{-1}=x$$
Use te reccurence relations to find $I_{n}$ as a function of "x" and "n".
Daniel.

I understand this. This is reminding me of my discrete math course. From what I remember, I learned how to find an explicit formula given the recurrence relation but that was only for 1st or 2nd order, homogeneous, linear, and constant co-efficient equations. Now this is calculus and I'm confused since we have these variables and ln x. I don't know how to go about solving this. Could you show me how to solve this recurrence relation?

 

[phoenixthoth]

$$I_{n}=x(\ln x)^{n}-nI_{n-1}$$ and $$I_{0}=x$$

Calulate $$I_{1}$$ using the first formula with $$n=1$$ and $$I_{0}=x$$. Then calculate the next, and next, and next...until you see the pattern. Then, if required, prove your formula one way or another.