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Category: Integration

a-line-L-intersects-0-0-and-the-curve-y-x-2-at-x-t-What-is-the-equation-of-the-line-What-is-the-area-between-L-and-y-from-x-0-to-x-t-

Question Number 8070 by FilupSmith last updated on 29/Sep/16 $$\mathrm{a}\:\mathrm{line}\:{L}\:\mathrm{intersec}{ts}\:\left(\mathrm{0},\:\mathrm{0}\right)\:{and}\:{the}\:{curve} \\ $$$${y}={x}^{\mathrm{2}} \:\:\:{at}\:{x}={t}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{line}? \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{area}\:\mathrm{between}\:{L}\:\mathrm{and}\:{y}\:\mathrm{from} \\ $$$${x}=\mathrm{0}\:\mathrm{to}\:{x}={t}? \\ $$ Answered by sandy_suhendra last…

nice-calculus-0-1-0-1-1-x-1-xy-ln-xy-2019-dxdy-

Question Number 139101 by mnjuly1970 last updated on 22/Apr/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:…….{nice}\:\:\:{calculus}….. \\ $$$$\boldsymbol{\phi}\overset{???} {=}\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{1}−{x}}{\mathrm{1}−{xy}}\left(−{ln}\left({xy}\right)\right)^{\mathrm{2019}} {dxdy} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:……… \\ $$ Answered by Dwaipayan…

evaluate-lnx-dx-

Question Number 73545 by Rio Michael last updated on 13/Nov/19 $${evaluate}\:\:\int{lnx}\:{dx} \\ $$ Commented by Tony Lin last updated on 13/Nov/19 $${integration}\:{by}\:{part} \\ $$$$\int{f}\:'\left({x}\right){g}\left({x}\right)={f}\left({x}\right){g}\left({x}\right)−\int{f}\left({x}\right){g}\:'\left({x}\right) \\…

4-x-4-x-dx-

Question Number 7960 by tawakalitu last updated on 25/Sep/16 $$\int\frac{\mathrm{4}\:−\:{x}}{\mathrm{4}\:+\:{x}}\:{dx} \\ $$ Commented by Yozzia last updated on 25/Sep/16 $$\int\frac{\mathrm{4}−{x}}{\mathrm{4}+{x}}{dx}=\int\left(\frac{\mathrm{8}}{\mathrm{4}+{x}}−\mathrm{1}\right){dx}=\mathrm{8}{ln}\left({x}+\mathrm{4}\right)−{x}+{c} \\ $$ Commented by tawakalitu…

5-x-1-x-4-dx-

Question Number 7961 by tawakalitu last updated on 25/Sep/16 $$\int\frac{\mathrm{5}\:−\:{x}}{\mathrm{1}\:+\:\sqrt{{x}\:−\:\mathrm{4}}}\:{dx} \\ $$ Answered by Yozzia last updated on 25/Sep/16 $${I}=\int\frac{\mathrm{5}−{x}}{\mathrm{1}+\sqrt{{x}−\mathrm{4}}}{dx} \\ $$$${Let}\:{t}=\sqrt{{x}−\mathrm{4}}\Rightarrow{dt}=\frac{\mathrm{1}}{\mathrm{2}}\left({x}−\mathrm{4}\right)^{−\mathrm{1}/\mathrm{2}} {dx} \\ $$$${dx}=\mathrm{2}\left({x}−\mathrm{4}\right)^{\mathrm{1}/\mathrm{2}}…

sin-1-x-1-x-2-dx-

Question Number 7959 by tawakalitu last updated on 25/Sep/16 $$\int\frac{{sin}^{−\mathrm{1}} \left({x}\right)}{\mathrm{1}\:−\:{x}^{\mathrm{2}} }\:{dx} \\ $$ Commented by nick last updated on 26/Sep/16 $${do}\:{you}\:{mean}\:\int\frac{\mathrm{sin}^{−\mathrm{1}} \left({x}\right)}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx} \\…