This is a simple calculus question, but it's been a while since I've had to use partial fraction decomposition. I need to integrate the following:
$$\int_{0}^\infty \frac{xdx}{(x+\theta)^4} $$
I approached this via partial fractions and used the following decomposition:
$$ \frac{x}{(x+\theta)^4} = \frac{A}{x+\theta} + \frac{B}{(x+\theta)^2} +\frac{C}{(x+\theta)^3} + \frac{D}{(x+\theta)^4} $$
which led to
$$ x = (A)x^3 + (A\theta +2A\theta +B)x^2 + (A2\theta^2 + A \theta^2 +2\theta +C)x + (B\theta^2 + D + A\theta^3) $$
then equating coefficients yields the following system
$$A = 0 \\ A\theta +2A\theta + B = 0 \\ 2A\theta^2 + A \theta^2 + 2\theta + C = 1 \\ B\theta^2 + D + A\theta^3 = 0$$
resulting in the following values:
$$ A=0\\ B=0\\ C=1-2\theta\\ D=0$$
and plugging these back in gives
$$ \frac{x}{(x+\theta)^4} = \frac{1-2\theta}{(x+\theta)^3}$$
But, Wolfram is telling me that
$$ \frac{x}{(x+\theta)^4} = \frac{1}{(x+\theta)^3} - \frac{\theta}{(x+\theta)^4}.$$
I thought that maybe these would yield the same result, but they definitely do not. I've double-checked my work many times here and can't find my mistake. If anyone could help me out here, I'd appreciate. Or, of course, if anyone knows a simpler way to tackle this integral, that'd obviously be appreciated, as well. Thanks!
EDIT: Yes, of course, a simple substitution suffices here. Sheesh... that's a little embarrassing. Thanks, everyone. Still, I am curious as to what went wrong with my partial fraction decomposition...