I am not getting the right answer with alternate completing the square method for the integral
$$\int\frac{x^2}{(3+4x-4x^2)^{3/2}}dx$$
I've looked up how to do this problem and when they complete the square it's using the $(\frac{-b}{a})^2$ method where, in the denominator, they factor out the $4$, and then make:
$$x^2-x=x^2-x+{1\over4}$$ and then(
$$x^2-x+{1\over4}=(x-\frac{1}{2})^2$$
which I understand, but is not the first option that popped into my head.
What I don't understand is why all the tutorials I've looked up are not completing the square in the traditional method. Here's my work: denominator:
$$3+4x-4x^2$$
$$(-4x^2+4x-1)+4$$
$$-(4x^2-4x+1)+4$$
$$-(2x-1)^2+4$$
So the integral is now:
$$\int\frac{x^2}{(4-u^2)^\frac{3}{2}}$$
So I do the rest of the work by using trig substitution now:
$u=2x-1$
$x=\frac{u+1}{2}$
Now I subbed in the trig identities:
$u=2\sin\theta$
$du=2\cos\theta d\theta$
$$\int\frac{(\frac{u+1}{2})^22\cos\theta}{\sqrt{4-4\sin^2\theta}^3}d\theta$$
$$\int\frac{\frac{(2\sin\theta+1)^2}{4}2\cos\theta}{\sqrt{4\cos^2\theta}^3}d\theta$$
$$\int\frac{\frac{(2\sin\theta+1)^2}{4}2\cos\theta}{(2\cos\theta)^3}d\theta$$
$$\int\frac{\frac{(2\sin\theta+1)^2}{4}2\cos\theta}{8\cos^3\theta}d\theta$$
$$\int\frac{\frac{(4\sin^2\theta+4\sin\theta+1)}{4}2\cos\theta}{8\cos^3\theta}d\theta$$
$$\int\frac{(\frac{4\sin^2\theta}{4}+\frac{4\sin\theta}{4}+\frac{1}{4})2\cos\theta}{8\cos^3\theta}d\theta$$
$$\int\frac{(\sin^2\theta+\sin\theta+\frac{1}{4})2\cos\theta}{8\cos^3\theta}d\theta$$
$$\int\frac{(\sin^2\theta+\sin\theta+\frac{1}{4})}{8\cos^3\theta}*\frac{2\cos\theta}{1}d\theta$$
$$\int\frac{(\sin^2\theta+\sin\theta+\frac{1}{4})}{4\cos^2\theta}d\theta$$
$$\int\frac{\sin^2\theta}{4\cos^2\theta}d\theta+\int\frac{\sin\theta}{4\cos^2\theta}d\theta+\int\frac{\frac{1}{4}}{4\cos^2\theta}d\theta$$
$$\frac{1}{4}\int \tan^2\theta d\theta+\frac{1}{4}\int\frac{\sin\theta}{\cos^2\theta}d\theta+\int\frac{1}{16\cos^2\theta}d\theta$$
For the second integral, I subbed $u=\cos\theta$ and $-du=\sin\theta d\theta$
$$\frac{1}{4}\int \sec^2\theta-1d\theta-\frac{1}{4}\int\frac{du}{u^2}+\frac{1}{16}\int \sec^2\theta d\theta$$
$$[\frac{\tan\theta}{4}-\frac{\theta}{4}]+[\frac{1}{4\cos\theta}]+[\frac{\tan\theta}{16}]+C$$
On a right triangle where $\sin\theta=\frac{2x-1}{2}$ and $\cos\theta=\frac{\sqrt{4-(2x-1)^2}}{2}$, I subbed these values back in to make the integral in terms of $x$:
$$\frac{2x-1}{4\sqrt{4-(2x-1)^2}}-\frac{\arcsin(\frac{2x-1}{2})}{4}+\frac{1}{4(\frac{\sqrt{4-(2x-1)^2}}{2})}+\frac{2x-1}{16\sqrt{4-(2x-1)^2}}+C$$
This does not seem to be the right answer when I compare it to a calculator answer, so I'm guessing I either completed the square wrong or made an algebraic mistake? I've done this problem three or four times now and am getting a little frustrated over what I'm guessing is a easy fix. If anyone can help, I'd really appreciate it.
Best Answer
Except some minor mistakes, your solution is correct but you led to wrong answer.
Mistake 1
In the first integral after substituting $u=\sin \theta$ you used $dx=du$ while $dx={1\over 2}du$.
Mistake 2
From $2$nd to $3$rd integral, you used $|\cos \theta|=\cos \theta$ while you should have dealt with $|\cos \theta|$ till the result.
Fixing up
Fixing these two mistakes, the final integral to be solved would become$$\int\frac{\sin^2\theta}{8\cos\theta|\cos\theta|}d\theta+\int\frac{\sin\theta}{8\cos\theta|\cos\theta|}d\theta+\int\frac{1}{32\cos\theta|\cos\theta|}d\theta\\={4+5\sin\theta-4\theta\cos\theta\over 32|\cos\theta|}+C$$