Question

Integration!
\[\int \frac{x+3}{\sqrt{x+6}}dx\]

Answer 1

So lemme see.
\(u = x + 6\)
\(du = dx\)
And that means \(x + 3 = u - 3\)
\[\int \frac{u-3}{\sqrt{u}}\]
So far so good?

Answer 2

ok

Answer 3

next?

Answer 4

\[\int \frac{u}{\sqrt{u}} - \frac{3}{\sqrt{u}}\]\[\int u^{1/2} - 3u^{-1/2}\]\[\frac{2}{3}u^{3/2} - 6u^{1/2} + C\]

Answer 5

next?

Answer 6

I guess it's just\[\frac{2}{3}(x + 6)^{3/2} - 6(x + 6)^{1/2} + C\]Can I clean that up any?

Answer 7

2(x+6)^1/2 (1/3 (x+6)^3-3) +C
how ugly it is , hahahaha

Answer 8

It's a good idea though. Maybe factor something out.
\[\sqrt{x + 6}\left(\frac{2}{3}(x + 6) - 6\right) + C\]\[\sqrt{x + 6}\left(\frac{2x}{3} - 2\right) + C\]\[\frac{2}{3}\sqrt{x + 6}(x - 3) + C\]
Well that looks nicer.