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CalculusEasyNon-Elementary Integration

Non-Elementary Integration · Easy problem

Frullani's theorem says that if ff is continuous, f(0)f(0) exists and f(x)→f(∞)f(x)\to f(\infty) as x→∞x\to\infty, then ∫0∞f(ax)−f(bx)x dx=(f(0)−f(∞))ln⁡ba\displaystyle\int_0^{\infty}\dfrac{f(ax)-f(bx)}{x}\,dx=\bigl(f(0)-f(\infty)\bigr)\ln\dfrac{b}{a} for a,b>0a,b>0. Evaluate ∫0∞e−2x−e−5xx dx\displaystyle\int_0^{\infty}\dfrac{e^{-2x}-e^{-5x}}{x}\,dx.