미적분학(Calculus) 잡다한 문제 1

Find the value of the constant $C$ for which the integral

$$\int^{\infty}_0 \left(\frac{x}{x^2+1}-\frac{C}{3x+1}\right)dx$$

converges. Evaluate the integral for value of $C$

Sol)

$$\lim_{t \to \infty} \int^t_0 \left(\frac{x}{x^2+1}-\frac{C}{3x+1}\right)dx$$

$$=\lim_{t \to \infty} \left(\int^t_0 \frac{x}{x^2+1}dx-\int^t_0 \frac{C}{3x+1}dx\right)$$

$$=\lim_{t \to \infty} \left\{\left[\frac{1}{2}\ln\vert{x^2+1}\vert\right]^t_0-\left[\frac{C}{3}\ln\vert{3x+1}\vert\right]^t_0\right\}$$

$$=\lim_{t \to \infty} \left\{\frac{1}{2}\ln(t^2+1)-\frac{C}{3}\ln(3t+1)\right\}$$

$$=\lim_{t \to \infty} \left\{\ln{(t^2+1)^\frac{1}{2}}-\ln{(3t+1)^\frac{C}{3}}\right\}$$

$$=\lim_{t \to \infty} \left\{\ln\frac{(t^2+1)^\frac{1}{2}}{(3t+1)^\frac{C}{3}}\right\}$$

여기서 $t$가 $\infty$로 가므로 $\ln$속 분모, 분자의 최고차항의 계수가 같아야 한다.

$$2\cdot\frac{1}{2}=1\cdot\frac{c}{3}$$

$$\therefore \; C=3$$