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Category: Integration

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

Question Number 8762 by tawakalitu last updated on 26/Oct/16 $$\int\mathrm{x}^{\mathrm{2}} \left(\mathrm{2x}\:+\:\mathrm{1}\right)^{\mathrm{1}/\mathrm{2}} \:\mathrm{dx} \\ $$ Answered by sou1618 last updated on 26/Oct/16 $${I}=\int{x}^{\mathrm{2}} \sqrt{\mathrm{2}{x}+\mathrm{1}}{dx} \\ $$$${t}=\mathrm{2}{x}+\mathrm{1}…

x-3x-1-dx-

Question Number 8763 by tawakalitu last updated on 26/Oct/16 $$\int\mathrm{x}\sqrt{\mathrm{3x}\:+\:\mathrm{1}}\:\:\mathrm{dx} \\ $$ Commented by FilupSmith last updated on 26/Oct/16 $${u}=\mathrm{3}{x}+\mathrm{2}\Rightarrow{x}=\frac{\mathrm{1}}{\mathrm{3}}\left({u}−\mathrm{2}\right) \\ $$$${du}=\mathrm{3}{dx} \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}\int\mathrm{3}{x}\sqrt{\mathrm{3}{x}+\mathrm{2}}{dx} \\…

n-0-sin-n-1-x-4-n-1-

Question Number 139826 by qaz last updated on 01/May/21 $$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{sin}\:\left[\left({n}−\mathrm{1}\right){x}\right]}{\mathrm{4}^{{n}+\mathrm{1}} }=? \\ $$ Answered by mnjuly1970 last updated on 01/May/21 $$\:\:\:\:\:\Omega:=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{sin}\left(\left({n}−\mathrm{1}\right){x}\right)}{\mathrm{4}^{{n}+\mathrm{1}}…

0-1-x-ln-1-1-x-dx-solution-0-1-x-n-1-x-n-n-dx-n-1-1-n-0-1-x-n-1-2-dx-n-1-1-n-n-3-2-

Question Number 139811 by mnjuly1970 last updated on 01/May/21 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \sqrt{{x}}\:{ln}\left(\frac{\mathrm{1}}{\mathrm{1}−{x}}\right){dx} \\ $$$$\:\:\:\:\:{solution}: \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\phi}:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \sqrt{{x}}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{x}^{{n}} }{{n}}\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\::=\underset{{n}=\mathrm{1}}…

Evaluate-the-integral-a-b-a-b-a-a-b-a-2-a-a-b-b-a-2-dt-In-which-a-b-are-the-derivatives-of-a-b-with-respect-to-t-

Question Number 8734 by tawakalitu last updated on 24/Oct/16 $$\mathrm{Evaluate}\:\mathrm{the}\:\mathrm{integral} \\ $$$$\int\left[\mathrm{a}\left(\overset{\bullet} {\mathrm{b}}.\mathrm{a}\:+\:\mathrm{b}.\overset{\bullet} {\mathrm{a}}\right)\:+\:\overset{\bullet} {\mathrm{a}}\left(\mathrm{b}.\mathrm{a}\right)\:−\:\mathrm{2}\left(\overset{\bullet} {\mathrm{a}}.\mathrm{a}\right)\mathrm{b}\:−\:\overset{\bullet} {\mathrm{b}}\mid\mathrm{a}\mid^{\mathrm{2}} \right]\mathrm{dt} \\ $$$$\mathrm{In}\:\mathrm{which}\:\:\overset{\bullet} {\mathrm{a}}\:,\overset{\bullet} {\mathrm{b}}\:\:\mathrm{are}\:\mathrm{the}\:\mathrm{derivatives}\:\mathrm{of}\:\:\mathrm{a},\mathrm{b}\:\mathrm{with}\: \\ $$$$\mathrm{respect}\:\mathrm{to}\:\mathrm{t} \\…