MathematicsMedium64×since 2004Q3997Let I(x)=∫x2(xsec2x+tanx)(xtanx+1)2dxI(x)=\int \frac{x^{2}\left(x \sec ^{2} x+\tan x\right)}{(x \tan x+1)^{2}} d xI(x)=∫(xtanx+1)2x2(xsec2x+tanx)dx. If I(0)=0I(0)=0I(0)=0, then I(π4)I\left(\frac{\pi}{4}\right)I(4π) is equal to :Aloge(π+4)232−π24(π+4)\log _{e} \frac{(\pi+4)^{2}}{32}-\frac{\pi^{2}}{4(\pi+4)}loge32(π+4)2−4(π+4)π2Bloge(π+4)216−π24(π+4)\log _{e} \frac{(\pi+4)^{2}}{16}-\frac{\pi^{2}}{4(\pi+4)}loge16(π+4)2−4(π+4)π2Cloge(π+4)216+π24(π+4)\log _{e} \frac{(\pi+4)^{2}}{16}+\frac{\pi^{2}}{4(\pi+4)}loge16(π+4)2+4(π+4)π2Dloge(π+4)232+π24(π+4)\log _{e} \frac{(\pi+4)^{2}}{32}+\frac{\pi^{2}}{4(\pi+4)}loge32(π+4)2+4(π+4)π2Check answerSkip