題目 5. (10 points) Evaluate ∫ln14ln12ex1−4e2x dx.\int_{\ln \frac{1}{4}}^{\ln \frac{1}{2}} \frac{e^x}{\sqrt{1-4e^{2x}}}\,dx.∫ln41ln211−4e2xexdx. 解答 解法一 令 u=2exu=2e^xu=2ex，則 ∫ln(1/4)ln(1/2)ex1−4e2x dx=12∫1/21du1−u2=π6.\int_{\ln(1/4)}^{\ln(1/2)} \frac{e^x}{\sqrt{1-4e^{2x}}}\,\mathrm dx =\frac12\int_{1/2}^1\frac{\mathrm du}{\sqrt{1-u^2}} =\boxed{\frac\pi6}.∫ln(1/4)ln(1/2)1−4e2xexdx=21∫1/211−u2du=6π.
