Let f(x) = (2x)/((x^2+1)(x^2+3)),dx.

Question

Let $f(x) = \displaystyle\int\frac{2x}{(x^2+1)(x^2+3)}\,dx$. If $f(3)=\dfrac12(\log_e5-\log_e6)$, then $f(4)$ is equal to:

Accepted Answer

Let $x^2=t$: $\displaystyle\int\frac{dt}{(t+1)(t+3)} = \frac12\int\left(\frac1{t+1}-\frac1{t+3}\right)dt$

$$f(x) = \frac12\ln\left|\frac{x^2+1}{x^2+3}\right|+C$$

From $f(3)=\dfrac12(\ln10-\ln12)+C = \dfrac12(\ln5-\ln6) \implies C=0$ (since $\ln(10/12)=\ln(5/6)$).

$$f(4) = \frac12\ln\frac{17}{19} = \frac12(\log_e17-\log_e19)$$

Solutions · 1