# Question and Answers Forum

Integration Questions

Question Number 93248 by i jagooll last updated on 12/May/20

$$\int\underset{\mathrm{0}} {\overset{\mathrm{1}} {\:}}\:\mathrm{ln}\left(\mathrm{x}\right)\:\mathrm{dx}\: \\$$

Commented byabdomathmax last updated on 12/May/20

$$\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}\right){dx}\:={lim}_{{a}\rightarrow\mathrm{0}^{+} } \:\:\int_{{a}} ^{\mathrm{1}} {ln}\left({x}\right){dx} \\$$ $$={lim}_{{a}\rightarrow\mathrm{0}^{+} } \:\:\:\:\left[{xlnx}−{x}\right]_{{a}} ^{\mathrm{1}} \: \\$$ $$={lim}_{{a}\rightarrow\mathrm{0}^{+} } \:\:\left(−\mathrm{1}−{alna}+{a}\right)\:=−\mathrm{1} \\$$

Answered by john santu last updated on 12/May/20

$$\mathrm{D}.\mathrm{I}\:\mathrm{method}\: \\$$ $$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\mathrm{ln}\left({x}\right)\:{dx}\:=\:\underset{{p}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\left\{\:\underset{\mathrm{p}} {\overset{\mathrm{1}} {\int}}\:\mathrm{ln}\left({x}\right)\:{dx}\:\right\} \\$$ $$\left.=\:\underset{{p}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\left\{\:{x}\:\mathrm{ln}\left({x}\right)−{x}\:\right\}\right]_{{p}} ^{\mathrm{1}} \\$$ $$=\:\underset{{p}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\left\{−\mathrm{1}+{p}−{p}\mathrm{ln}\left({p}\right)\right\} \\$$ $$=\:−\mathrm{1}+\:\underset{\mathrm{p}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:{p}\left(\mathrm{1}−\mathrm{ln}\left({p}\right)\right) \\$$ $$=\:−\mathrm{1}+\:\underset{{p}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{ln}\left({p}\right)}{\left(\frac{\mathrm{1}}{{p}}\right)} \\$$ $$=−\mathrm{1}+\:\underset{{p}\rightarrow\mathrm{0}^{+} } {\mathrm{li}\overset{{L}\mathrm{H}} {\mathrm{m}}}\:\left(\frac{−\left(\frac{\mathrm{1}}{{p}}\right)}{−\left(\frac{\mathrm{1}}{{p}^{\mathrm{2}} }\right)}\right)\: \\$$ $$=\:−\mathrm{1}\:+\:\underset{{p}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\left({p}\right)\:=\:−\mathrm{1}+\mathrm{0}\:=\:−\mathrm{1} \\$$

Commented byi jagooll last updated on 12/May/20

$$\mathrm{thank}\:\mathrm{you}... \\$$