Find the value $C$ such that $\int^{\infty}_{0}(\frac{x}{x^2+1} - \frac{C}{3x+1})dx$ converges
$$\int^{\infty}_{0}(\frac{x}{x^2+1} - \frac{C}{3x+1})dx$$
How do I do this?
The only thing I can think of is creating a common denominator?
$$\displaystyle\int\limits^{\infty}_{0} \dfrac{x^2\left(3-c\right)+x-c}{\left(x^2+1\right)\left(3x+1\right)}\,\mathrm{d}x$$
Something tells me $C = 3$ is right, but I don't know why I think that or if it's even right.
A quotient to infinity will converge if its denominator is more than one degree higher than it's numerator.
If C is anything other than 3, then the denominator is exactly one degree higher and thus the integral is not convergent. If C is exactly 3, then the denominator is two degrees higher and the integral is convergent.