Need help with difficult integration by parts problem

[kkidd002]

Homework Statement

integrate: (x^3) e^(3x^2) dx

Homework Equations

uv- integral vdu

The Attempt at a Solution

i've tried this many times on paper and can't get the right answer. I'm starting to get really frustrated. Please help, and be specific as possible.

 

[bob1182006]

Since you haven't shown any work I'll assume the worst :s.

First you should make a substitution if possible, to get just e^t.

And when you have an integral of the form:

$$\int x^n T(x) dx$$
where T(x) is a Transcendental function (e^x, sin x, cos x, a^[bx+c], ln x, etc..). And it's antiderivative should be easier than the entire function, otherwise you'll go from a hard integral to a harder one.

to solve that type of integral you need to make the substitutions:
$$u=x^n$$
$$v'=T(x)$$
and you will need to integrate n times, continue with u=x^n and eventually you will get rid of the x term and have only T(x) which would be a trivial integral. ex. $$...-\int cos x dx$$
Also when integrating and say you have $$-3\int x^n cos 3x dx$$ don't just choose u=x^n; v'=cos 3x, place that coefficient on either u or v' to maybe cancel it out. A better choice would be u=x^n; v'=-3cos 3x, which gives u'=nx^(n-1); v=-sin3x
This way you don't have to worry about multiplying everything by that - sign, you might forget it and get the entire thing wrong :/

 

[rock.freak667]

My advice is to rewrite the integral as this

$$\frac{1}{6}\int x^2(6xe^{3x^2}) dx$$

and then use integration by parts

Here is a nice hint to help you

$$\frac{d}{dx}(e^{3x^2})=6xe^{3x^2}$$

 

[transgalactic]

this is my solution

i added a file with my solution

 

[bob1182006]

could you just write it out? usually it takes a while before attachments are approved.

 

[transgalactic]

i don't know how to write integral signs like you do

 

[bob1182006]

click on the integral and you'll see the code it should be like this:

[ tex] \int ... x^2 [ /tex]

without the spaces before the tex and /tex

 

[transgalactic]

wow its a sintax languege like programming
i'll try to get used to it
about the attachments

it always instantly appears on my pc that my attachment massage
is added to the thread

 

[bob1182006]

it has to be approved by a moderator before we can see it I think.

and here's a thread that has guides on how to use the LaTeX typesetting that's really useful here.

 

[transgalactic]

ok thanks

 

[Gib Z]

My advice is to rewrite the integral as this

$$\frac{1}{6}\int x^2(6xe^{3x^2}) dx$$

and then use integration by parts

Here is a nice hint to help you

$$\frac{d}{dx}(e^{3x^2})=6xe^{3x^2}$$

Best solution here =]

 

[ebonhawkabc]

even easier is to separate x^3 into x * x^2 and then set u = x^2. then use integration by parts

 

[Ray Vickson]

You can do it in ASCII: you want int x^3*exp(3*x^2) dx. This looks nicer in 'tex', but is perfectly legible without it. If you wanted the definite integral from x=a to x=b you could write int(f(x) dx, x=a..b) or int_{x=a..b} f(x) dx.

RGV

 

[SammyS]

Hey ! This thread is nearly 4 years old! LOL.

 

[rock.freak667]

For the record, I still like my hint :rofl: