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Category: Integration

Question-59893

Question Number 59893 by aliesam last updated on 15/May/19 Commented by maxmathsup by imad last updated on 15/May/19 $${method}\:{using}\:{the}\:{formuae}\:\int_{\mathrm{0}} ^{\infty} \frac{{t}^{{a}−\mathrm{1}} }{\mathrm{1}+{t}}{dt}\:=\frac{\pi}{{sin}\left(\pi{a}\right)}\:\:{if}\:\mathrm{0}<{a}<\mathrm{1} \\ $$$${let}\:{I}\:=\int_{−\infty} ^{+\infty}…

0-100-dx-x-100-x-

Question Number 125421 by bemath last updated on 11/Dec/20 $$\:\underset{\mathrm{0}} {\overset{\mathrm{100}} {\int}}\:\frac{{dx}}{\:\sqrt{{x}\left(\mathrm{100}−{x}\right)}}\:?\: \\ $$ Answered by liberty last updated on 11/Dec/20 $${I}=\underset{\mathrm{0}} {\overset{\mathrm{100}} {\int}}\:\frac{{dx}}{\:\sqrt{{x}}\:\sqrt{\mathrm{100}−{x}}}\:;\:{let}\:\sqrt{{x}}\:=\:\mathrm{10}\:\mathrm{sin}\:{t}\: \\…

dx-x-x-x-2-

Question Number 125416 by bemath last updated on 10/Dec/20 $$\int\:\frac{{dx}}{\:\sqrt{{x}\sqrt{{x}}−{x}^{\mathrm{2}} }}\:? \\ $$ Answered by Dwaipayan Shikari last updated on 10/Dec/20 $$\int\frac{{dx}}{\:\sqrt{{x}}\sqrt{\sqrt{{x}}−{x}}}\:\:\:\:\:\:\:\:\:\:\sqrt{{x}}={t}\Rightarrow\frac{\mathrm{1}}{\mathrm{2}\sqrt{{x}}}=\frac{{dt}}{{dx}} \\ $$$$=\mathrm{2}\int\frac{{dt}}{\:\sqrt{{t}−{t}^{\mathrm{2}} }}=\mathrm{2}\int\frac{\mathrm{1}}{\:\sqrt{{t}}\left(\sqrt{\mathrm{1}−{t}}\right)}{dt}\:\:\:\:\:\:\:\:\:\:\:\:\:{t}={u}^{\mathrm{2}}…

The-parametric-equation-of-a-curve-are-x-3t-2-and-y-3t-t-2-Find-the-volume-generated-when-the-plane-bounded-by-the-curve-the-x-axis-and-the-ordinates-corresponding-to-t-0-and-t-2-rotates-abo

Question Number 190940 by Spillover last updated on 14/Apr/23 $$\mathrm{The}\:\mathrm{parametric}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{a}\:\mathrm{curve}\:\:\mathrm{are} \\ $$$$\mathrm{x}=\mathrm{3t}^{\mathrm{2}} \:\mathrm{and}\:\mathrm{y}=\mathrm{3t}−\mathrm{t}^{\mathrm{2}} . \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{generated} \\ $$$$\mathrm{when}\:\mathrm{the}\:\mathrm{plane}\:\mathrm{bounded}\:\mathrm{by}\:\mathrm{the}\:\mathrm{curve} \\ $$$$,\mathrm{the}\:\mathrm{x}−\mathrm{axis}\:\mathrm{and}\:\mathrm{the}\:\mathrm{ordinates}\: \\ $$$$\mathrm{corresponding}\:\mathrm{to}\: \\ $$$$\mathrm{t}=\mathrm{0}\:\:\:\mathrm{and}\:\mathrm{t}=\mathrm{2}\:\:\mathrm{rotates}\:\mathrm{about}\:\mathrm{the}\:\mathrm{y}−\mathrm{axis} \\…

Show-that-sech-x-tanh-x-x-2-cosh-x-

Question Number 190937 by Spillover last updated on 14/Apr/23 $$\mathrm{Show}\:\:\mathrm{that}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\:\frac{\mathrm{sech}\:\sqrt{\mathrm{x}}\:\mathrm{tanh}\:\sqrt{\mathrm{x}}}{\:\sqrt{\mathrm{x}}}=−\frac{\mathrm{2}}{\mathrm{cosh}\:\sqrt{\mathrm{x}}} \\ $$ Answered by ARUNG_Brandon_MBU last updated on 15/Apr/23 $${I}=\int\frac{\left(\mathrm{sech}\sqrt{{x}}\right)\left(\mathrm{tanh}\sqrt{{x}}\right)}{\:\sqrt{{x}}}{dx}=\int\frac{\mathrm{sinh}\sqrt{{x}}}{\:\sqrt{{x}}\mathrm{cosh}^{\mathrm{2}} \sqrt{{x}}}{dx} \\ $$$${t}=\mathrm{cosh}\sqrt{{x}}\:\Rightarrow{dt}=\frac{\mathrm{sinh}\sqrt{{x}}}{\mathrm{2}\sqrt{{x}}}{dx}…