heres the integral
x is between 0 and 2
y is between 0 and 5x
I need to find the exact value of the double integral 5x^2e^(xy)
I'm not really sure how to do this so any help would be greatly appreciated. Thanks in advance.
heres the integral
x is between 0 and 2
y is between 0 and 5x
I need to find the exact value of the double integral 5x^2e^(xy)
I'm not really sure how to do this so any help would be greatly appreciated. Thanks in advance.
$\displaystyle
\int_0^2\int_0^{5x}5x^2e^{xy}\,dy\,dx
$
$\displaystyle
=\int_0^2 5x^2 [\int_0^{5x}e^{xy}\,dy]\,dx
$
$\displaystyle
=\int_0^2 5x^2 [\frac{e^{xy}}{x}]|_0^{5x}\,dx
$
$\displaystyle
=\int_0^2 5x [e^{xy}]|_0^{5x}\,dx
$
$\displaystyle
=\int_0^2 5x (e^{5x^2}-1)\,dx
$
$\displaystyle
=\int_0^2 5x e^{5x^2}\,dx - \int_0^2 5x \,dx
$
$\displaystyle
=\frac{1}{2}\int_0^2 e^{5x^2}\,d{(5x^2)} - \int_0^2 5x \,dx
$
Get the idea?
Thank you, what I didn't realize was to factor out the 5x^2, I was staring at the whole integral and I had no idea how to approach it I was thinming integration by parts or something but it just did not seem right.
just a few questions about the end?
first where does the 1/2 come from?
second how would I integrate that expression e^(5x^2)d(5x^2)? The other half of it I know, i just have a lot of trouble integrating expressions with e.
thanks again for helping me with this.