Differentiation Questions

Question Number 9113 by hmhjkk last updated on 19/Nov/16

$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{x}^{\mathrm{2}} /\mathrm{x}^{\mathrm{2}} +\mathrm{1}=\mathrm{0} \\$$

Commented by sou1618 last updated on 19/Nov/16

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} +\mathrm{1}}{dx}=\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{1}−\frac{\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{1}}{dx} \\$$$$\\$$$$\left({i}\right)\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{1}{dx}=\left[{x}\right]_{\mathrm{0}} ^{\mathrm{1}} =\mathrm{1} \\$$$$\\$$$$\left({ii}\right)\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{1}}{dx}\:\underset{{x}={tan}\theta} {\rightarrow}\:\int_{\mathrm{0}} ^{\pi/\mathrm{4}} \frac{\mathrm{1}}{{tan}^{\mathrm{2}} \theta+\mathrm{1}}×\left(\frac{{d}}{{d}\theta}{tan}\theta\right){d}\theta \\$$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\int_{\mathrm{0}} ^{\pi/\mathrm{4}} \frac{\mathrm{1}}{\left(\frac{{sin}^{\mathrm{2}} \theta+{cos}^{\mathrm{2}} \theta}{{cos}^{\mathrm{2}} \theta}\right)}×\frac{\mathrm{1}}{{cos}^{\mathrm{2}} \theta}{d}\theta \\$$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\int_{\mathrm{0}} ^{\pi/\mathrm{4}} {cos}^{\mathrm{2}} \theta×\frac{\mathrm{1}}{{cos}^{\mathrm{2}} \theta}{d}\theta \\$$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\left[\theta\right]_{\mathrm{0}} ^{\pi/\mathrm{4}} \\$$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\pi}{\mathrm{4}} \\$$$$\left(\left({i}\right),\left({ii}\right)\right)\Rightarrow \\$$$${ans}=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{4}}\pi \\$$