Integrate using Trigonometric Substitutions Evaluate the integral using trigonometric substitutions.  
$$\int{  x\over \sqrt{3-2x-x^2}} \,dx$$
I am familiar with using the right triangle diagram and theta, but I do not know which terms would go on the hypotenuse and sides in this case. If you can determine which numbers or $x$-values go on the hypotenuse, adjacent, and opposite sides, I can figure out the rest, although your final answer would help me check mine. Thanks!
 A: The "trick" in evaluating
$$\tag{1}
\int{x\over\sqrt{3-2x-x^2}}\,dx
$$
is to complete the square of the expression in the radicand: rewrite $3-2x-x^2$ as$$\tag{2}4-\color{maroon}{(x+1)}^2.$$
I'm not sure what this right triangle diagram you speak of is, but with the method I assume you're using, the second "trick" is to take advantage of one of the Pythagorean  Identities so that the square root in $(1)$ can be taken. Looking at $(2)$, you should be reminded of 
$$
a^2-\color{maroon}{a^2\sin^2\theta}=a^2\cos^2\theta.
$$
So, one may make the substitution $$\tag{3} (x+1)=2\sin\theta.$$
Then $$4-(x+1)^2 =4-(2\sin\theta)^2= 4-4\sin^2\theta= 4\cos^2\theta$$
Also, from our substitution rule $(3)$: $dx=2\cos\theta\, d\theta $ and $x=2\sin\theta-1$.
The integral $(1)$ then becomes
$$
\int { 2\sin\theta-1 \over2\cos\theta}\cdot2\cos\theta\,d\theta= \int(2\sin\theta-1)\,d\theta.
$$
I'll leave the rest for you... 
(and now I recall the triangle business: you can label two sides using  $\sin\theta=(x+1)/2$; usually, after you find the antiderivative and write back in terms of $x$, the triangle is used as an an aid to simplify your answer). 
A: $\int \frac{x}{\sqrt{4-(x+1)^2}}dx = \int \frac{2\sin\theta-1}{\sqrt{4-4\sin^2\theta}}(2\cos\theta)d\theta$
(using the substitution $x+1=2\sin\theta$)
$=\int\frac{2\sin\theta-1}{2\cos\theta}2\cos\theta d\theta$
$= \int (2\sin\theta-1) d\theta$
$=-2\cos\theta-\theta +C$
$=-2\left(\frac{\sqrt{3-2x-x^2}}{2}\right) - \sin^{-1}\left(\frac{x+1}{2}\right)+C$
$=-\sqrt{3-2x-x^2}- \sin^{-1}\left(\frac{x+1}{2}\right)+C$
