Evaluate the following double integral

[Piney1]

Homework Statement

Change the order of integration and evaluate the following double integral:

$$I = {\int_0^{1} \left({\int\limits_{y}^{1} 30 y\sqrt{1+x^3} \mathrm{d}x }\right) {\mathrm{d}y}$$


So thenn i did

$$= 30 \int_0^{1} \sqrt{1+x^3} \left({\int_0^{x} y \mathrm{d}y}\right) \mathrm{d}x$$

$$= 30 \int_0^{1} \sqrt{1+x^3} \left(\frac{x^2}{2} \right) \mathrm{d}x \end{align}$$

using integration by parts...

for $$\sqrt{1+x^3}$$
$$let u = \sqrt{1+x^3} \qquad du= \frac{1}{2} \left(\sqrt{1+x^3}\right) 3x^2 = \frac{3x^2}{2\sqrt{1+x^3}} \qquad dv = dx \qquad v = x$$

Thus!

$$= x \sqrt{1+x^3} - \int \frac{3x^3}{2\sqrt{1+x^3}} \mathrm{d}x$$


after that... i have no clue what to do. a lil help? thanks!
am i on the right track though?

 

[gabbagabbahey]

Why use IBP at all? What is $\frac{d}{dx}(1+x^3)^{3/2}$?

 

[Piney1]

don't we need to worry bout what's inside the bracket? when differentiating?

 

[gabbagabbahey]

Of course, use the chain rule.

 

[Piney1]

OHHHHHHHHH!
ahh dear.. i sure do love to make things complicated..

Thanks heaps! and there i was looking at tht question for hrs...