# Need help with difficult integration by parts problem

1. Oct 14, 2007

### kkidd002

1. The problem statement, all variables and given/known data

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

2. Relevant equations

uv- integral vdu

3. 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.

2. Oct 14, 2007

### 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 :/

3. Oct 14, 2007

### 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

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

4. Oct 14, 2007

### transgalactic

this is my solution

i added a file with my solution

#### Attached Files:

• ###### 4.JPG
File size:
11.7 KB
Views:
150
5. Oct 14, 2007

### bob1182006

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

6. Oct 14, 2007

### transgalactic

i dont know how to write integral signs like you do

7. Oct 14, 2007

### 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

8. Oct 14, 2007

### transgalactic

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

it always instantly appears on my pc that my attachment massage

9. Oct 14, 2007

### 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.

10. Oct 14, 2007

### transgalactic

ok thanks

11. Oct 15, 2007

### Gib Z

Best solution here =]

12. Jul 29, 2011

### ebonhawkabc

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

13. Jul 29, 2011

### 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

14. Jul 29, 2011

### SammyS

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

15. Jul 29, 2011

### rock.freak667

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