Solve the problem.
-Evaluate
$$\int _ { 0 } ^ { \infty } \frac { e ^ { - 2 x } - e ^ { - 8 x } } { x } d x$$
by writing the integrand as an integral.
A) $\ln \frac { 1 } { 4 }$
B) $\frac { \ln 2 } { \ln 8 }$
C) $\frac { \ln 8 } { \ln 2 }$
D) $\ln 4$

Question

Quizplus · Accepted Answer

The given integral can be evaluated by recognizing it as a form of the Frullani integral. The Frullani integral states that if $f(x)$ is a continuous function with $f(0)$ finite and $f(\infty) = 0$, then $\int_{0}^{\infty} \frac{f(ax) - f(bx)}{x} dx = \ln\left(\frac{b}{a}\right) f(0)$. Applying this to the given integral, where $f(x) = e^{-x}$, $a = 2$, and $b = 8$, we get $\ln\left(\frac{8}{2}\right) = \ln(4)$.

Solve the Problem ∫0∞e−2x−e−8xxdx\int _ { 0 } ^ { \infty } \frac { e ^ { - 2 x } - e ^ { - 8 x } } { x } d x∫0∞​xe−2x−e−8x​dx

Solve the Problem $\int _ { 0 } ^ { \infty } \frac { e ^ { - 2 x } - e ^ { - 8 x } } { x } d x$