Tích phân \(\int_0^{\frac{\pi}{4}}x\tan^2x\text{d}x\) bằng
\(-\dfrac{\pi^2}{32}+\dfrac{\pi}{4}+\ln\dfrac{\sqrt{2}}{2}\). \(\dfrac{\pi^2}{32}+\dfrac{\pi}{4}+\ln\dfrac{\sqrt{2}}{2}\). \(-\dfrac{\pi^2}{32}-\dfrac{\pi}{4}+\ln\dfrac{\sqrt{2}}{2}\). \(-\dfrac{\pi^2}{32}+\dfrac{\pi}{4}-\ln\dfrac{\sqrt{2}}{2}\). Hướng dẫn giải:Cách 1: Dùng MTCT.
Cách 2: \(\int_0^{\frac{\pi}{4}}x\tan^2x\text{d}x=\int^{\frac{\pi}{4}}_0x\left(1+\tan^2x-1\right)\text{d}x=\int^{\frac{\pi}{4}}_0x\text{d}\left(\tan x\right)-\int^{\frac{\pi}{4}}_0x\text{d}x\)
\(=x\tan x|^{\frac{\pi}{4}}_0-\int^{\frac{\pi}{4}}_0\tan x-\dfrac{x^2}{2}|^{\frac{\pi}{4}}_0=\dfrac{\pi}{4}+\int^{\frac{\pi}{4}}_0\dfrac{\text{d}\left(\cos x\right)}{\cos x}-\dfrac{\pi^2}{32}\)
\(=-\dfrac{\pi^2}{32}+\dfrac{\pi}{4}+\ln\left|\cos x\right||^{\frac{\pi}{4}}_0=-\dfrac{\pi^2}{32}+\dfrac{\pi}{4}+\ln\dfrac{\sqrt{2}}{2}\).