1. ## integration by parts

Am i doing something wrong? I cant seem to spot my mistake, the answer is in red (given by my teacher).

2. $\int x^3 \cdot e^{\frac{x^2}{2}} \, dx$

$t = \frac{x^2}{2}$

$dt = x \, dx$

$2 \int x \cdot \frac{x^2}{2} \cdot e^{\frac{x^2}{2}} \, dx$

substitute ...

$2 \int t \cdot e^{t} \, dt$

now do parts

3. Originally Posted by skeeter
$\int x^3 \cdot e^{\frac{x^2}{2}} \, dx$

$t = \frac{x^2}{2}$

$dt = x \, dx$

$2 \int x \cdot \frac{x^2}{2} \cdot e^{\frac{x^2}{2}} \, dx$

substitute ...

$2 \int t \cdot e^{t} \, dt$

now do parts
thanks! just wondering, where did i go wrong in my first answer?

4. Originally Posted by gomes
thanks! just wondering, where did i go wrong in my first answer?
$e^{\frac{x^2}{2}}$ does not have an elementary antiderivative ... so, it cannot be chosen as "dv"

5. let $u = x^{2}$

and $dv = x e^{x^{2}/2}$

6. While, in differentiating something like $e^{x^2}$ you can just multiply by the derivative of $x^2$, to get $2xe^{x^2}$, when integrating you cannot just divide by the derivative- it must already be in the integral.

7. thanks everyone, very helpful! erm, could someone clarify for me: how do i know if something has an elementary antiderivative or not?

8. Originally Posted by gomes
thanks everyone, very helpful! erm, could someone clarify for me: how do i know if something has an elementary antiderivative or not?
experience

http://www.math.unt.edu/integration_bee/AwfulTruth.html

9. Originally Posted by gomes
How do i know if something has an elementary antiderivative or not?
Differential Galois theory.

10. Thanks everyone!