1. Integral Question

Umm, I'm not quite sure how to ask this....

$
\int {x^3 e^{x^2 } } dx = \tfrac{1}
{2}\int {x^2 e^{x^2 } } d(x^2 )
$

How do you do that? What's going on here? As in changing dx to d(x^2). Is this a method I haven't been taught?....or do I not fully understand the concept of "with respect to x"? I see that d(x^2) equals 2x giving you the same integral but I need further clarification....

Thanks.

2. This is just an alternative notation for making a substitution in an integral. If you replace $x^2$ by $y$ in the right-hand integral then it becomes $\tfrac{1}{2}\!\!\int y e^y\,dy$, which exactly what you would get by making the substitution $y=x^2$ in the left-hand integral.

3. Originally Posted by RedBarchetta
Umm, I'm not quite sure how to ask this....

$
\int {x^3 e^{x^2 } } dx = \tfrac{1}
{2}\int {x^2 e^{x^2 } } d(x^2 )
$

How do you do that? What's going on here? As in changing dx to d(x^2). Is this a method I haven't been taught?....or do I not fully understand the concept of "with respect to x"? I see that d(x^2) equals 2x giving you the same integral but I need further clarification....

Thanks.
$d(x^2) = 2 x \, dx$. Substitute and you get your original integral.

Perhaps a more obvious approach for you is to make the substitution

$u = x^2 \Rightarrow \frac{du}{dx} = 2x \Rightarrow dx = \frac{du}{2x}$

$\frac{1}{2} \int u e^u \, du$ which can be solved using integration by parts.