$\displaystyle
\int (x^2)/(\sqrt(9+x^2)) dx
$
** Thats the best I can make this equation look...
Anyways I have been trying to get this thing for hours, I clearly do not know what I'm doing. Advice geatly apriciated!
Okay sooo I do all that and then I get to the part where it loops back to my original integral. But since i use integration by parts twice the two negatives make a positive and then you have somthing like:
A = original integral
A = stuff + A
so i clearly did somthing wrong...
Heres my step my step:
$\displaystyle
\int (x^2)/(\sqrt(9-x^2)) = 9\int (tan(x)^2sec(x) dx
$
$\displaystyle
9\int (tan(x)^2sec(x) dx = 9[tan(x)sec(x) - \int sec(x)^3
$
$\displaystyle
9[tan(x)sec(x) - \int sec(x)^3 = 9[tan(x)sec(x) - [sec(x)tan(x) - \int tan(x)^2sec(x)]]
$
$\displaystyle
9\int (tan(x)^2sec(x) dx = 9tan(x)sec(x) - 9sec(x)tan(x) + 9\int (tan(x)^2sec(x) dx
$
At this point its clearly wrong what did I do?
$\displaystyle \int \frac{x^2}{\sqrt{9+x^2}}\,dx$
Integration by parts:
Let $\displaystyle x=u$ and $\displaystyle dv=\frac{x}{\sqrt{9+x^2}}$
Thus, $\displaystyle dx=du$ and $\displaystyle v=\sqrt{9+x^2}$
Your integral is now: $\displaystyle x\sqrt{9+x^2}-\int \sqrt{9+x^2}\,dx$
Use $\displaystyle x=3tan(t)$ and $\displaystyle dx=3sec^2t$.
Now you have: $\displaystyle x\sqrt{9+x^2}-\int \sqrt{9+9tan^2t}*3sec^2t\,dt = x\sqrt{9+x^2}-9\int sec^3t\,dt$
Let $\displaystyle S=\int sec^3t\,dt$
Integration by parts again:
Let $\displaystyle u=sec(t)$ and $\displaystyle dv=sec^2(t)$.
Thus $\displaystyle du=sec(t)tan(t)$ and $\displaystyle v=tan(t)$ and the integral becomes: $\displaystyle sec(t)tan(t)-\int tan^2(t)sec(t)\,dt$.
This becomes: $\displaystyle sec(t)tan(t)-\int (sec^2(t)-1)sec(t)\,dt = sec(t)tan(t)-\int sec^3(t) + \int sec(t)\,dt$.
Thus $\displaystyle S = sec(t)tan(t) - S + \int sec(t)\,dt$ and $\displaystyle 2S = sec(t)tan(t) + \int sec(t)\,dt$.
Using a table of integrals, we find that $\displaystyle \int sec(t)\,dt = ln|sec(t)+tan(t)|$.
Thus $\displaystyle 2S = sec(t)tan(t) + ln|sec(t)+tan(t)|$.
Hence, $\displaystyle \int sec^3t\,dt = S = \frac{1}{2}sec(t)tan(t) + \frac{1}{2}ln|sec(t)+tan(t)|$.
Plugging $\displaystyle \int sec^3t\,dt$ back into the equation a few lines up, we have $\displaystyle x\sqrt{9+x^2}-9(\frac{1}{2}sec(t)tan(t) + \frac{1}{2}ln|sec(t)+tan(t)|)$.
Remember though, that since $\displaystyle x = 3tan(t)$, $\displaystyle t = tan^{-1}\frac{x}{3}$.
Thus, $\displaystyle x\sqrt{9+x^2}-9(\frac{1}{2}sec(t)tan(t) + \frac{1}{2}ln|sec(t)+tan(t)|)$ becomes $\displaystyle x\sqrt{9+x^2}-9(\frac{1}{2}sec(tan^{-1}\frac{x}{3})tan(tan^{-1}\frac{x}{3}) + \frac{1}{2}ln|sec(tan^{-1}\frac{x}{3})+tan(tan^{-1}\frac{x}{3})|)$.
Using the triangle, if the tangent of an angle is $\displaystyle \frac{x}{3}$, the secant of that angle is $\displaystyle \frac{3}{\sqrt{9+x^2}}$. Plugging everything makes that nasty equation become: $\displaystyle x\sqrt{9+x^2}-9(\frac{1}{2}*\frac{3}{\sqrt{9+x^2}}*\frac{x}{3}) + \frac{1}{2}ln|\frac{3}{\sqrt{9+x^2}}+\frac{x}{3}|)$.
Simplifying that yields: $\displaystyle x\sqrt{9+x^2}-\frac{9}{2}*\frac{x}{\sqrt{9+x^2}} + \frac{9}{2}ln|\frac{3}{\sqrt{9+x^2}}+\frac{x}{3}|$, which is what I think the answer is.
(You think that's ugly, you should see what it looks like in LaTex.)
I just realized I made a mistake at the end. When I said $\displaystyle sec(arctan(\frac{x}{3}))=\frac{3}{\sqrt{9+x^2}}$, I was wrong. It's actually $\displaystyle \frac{\sqrt{9+x^2}}{3}$, making the final answer:
$\displaystyle \frac{1}{2}x\sqrt{9+x^2}-\frac{9}{2}ln(\frac{1}{3}\sqrt{9+x^2}+\frac{1}{3}x )$
Sorry for the mistake.