Question Number 115892 by mohammad17 last updated on 29/Sep/20
Answered by Dwaipayan Shikari last updated on 29/Sep/20
![∫_0 ^4 xlogxdx [(x^2 /2)logx]_0 ^4 −[(x^2 /4)]_0 ^4 8log(4)−4](Q115914.png)
$$\int_{\mathrm{0}} ^{\mathrm{4}} \mathrm{xlogxdx} \\ $$$$\left[\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}\mathrm{logx}\right]_{\mathrm{0}} ^{\mathrm{4}} −\left[\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{4}}\right]_{\mathrm{0}} ^{\mathrm{4}} \\ $$$$\mathrm{8log}\left(\mathrm{4}\right)−\mathrm{4} \\ $$
Commented by mohammad17 last updated on 29/Sep/20
$${by}\:{simpson}\:{sir} \\ $$
Commented by mathmax by abdo last updated on 29/Sep/20
$$\mathrm{he}\:\mathrm{gives}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{by}\:\mathrm{pimpon}.... \\ $$
Commented by Dwaipayan Shikari last updated on 30/Sep/20
$$\int_{\mathrm{0}} ^{\mathrm{4}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\frac{\mathrm{4}−\mathrm{0}}{\mathrm{6}}\left(\mathrm{f}\left(\mathrm{0}\right)+\mathrm{4f}\left(\frac{\mathrm{4}}{\mathrm{2}}\right)+\mathrm{f}\left(\mathrm{b}\right)\right) \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{xlogx} \\ $$$$\mathrm{f}\left(\mathrm{0}\right)=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}xlogx}=\mathrm{x}\left(\mathrm{x}−\mathrm{1}\right)=\mathrm{0} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{4}} \mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{2}}{\mathrm{3}}\left(\mathrm{4}.\mathrm{2log}\left(\mathrm{2}\right)+\mathrm{4log4}\right)=\frac{\mathrm{32}}{\mathrm{3}}\mathrm{log}\left(\mathrm{2}\right)=\mathrm{7}.\mathrm{3935}.\:\:\left(\mathrm{approx}\right) \\ $$$$\mathrm{Actual}\:\mathrm{result}=\mathrm{16log}\left(\mathrm{2}\right)−\mathrm{4}=\mathrm{7}.\mathrm{090}.. \\ $$