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let2x-1-t-x-t-1-2-dx-dt-2-t-1-t-2dt-2-1-4-t-t-t-dt-continue-it-




Question Number 7819 by akmishra last updated on 17/Sep/16
let2x+1=t      x=(t−1)/2  dx=dt/2  ∫(t−1)(√t)/2dt/2  1/4∫(t(√t)−(√t))dt  continue.....it..
$${let}\mathrm{2}{x}+\mathrm{1}={t} \\ $$$$\:\:\:\:{x}=\left({t}−\mathrm{1}\right)/\mathrm{2} \\ $$$${dx}={dt}/\mathrm{2} \\ $$$$\int\left({t}−\mathrm{1}\right)\sqrt{{t}}/\mathrm{2}{dt}/\mathrm{2} \\ $$$$\mathrm{1}/\mathrm{4}\int\left({t}\sqrt{{t}}−\sqrt{{t}}\right){dt} \\ $$$${continue}…..{it}.. \\ $$$$\:\: \\ $$$$ \\ $$$$ \\ $$
Answered by prakash jain last updated on 02/Oct/16
1/4∫(t(√t)−(√t))dt  =(1/4)∫(t^(3/2) −t^(1/2) )dt  =(1/4)((t^(5/2) /(5/2))−(t^(3/2) /(3/2)))=(1/(10))t^(5/2) −(1/6)t^(3/2) +C
$$\mathrm{1}/\mathrm{4}\int\left({t}\sqrt{{t}}−\sqrt{{t}}\right){dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\int\left({t}^{\mathrm{3}/\mathrm{2}} −{t}^{\mathrm{1}/\mathrm{2}} \right){dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{{t}^{\mathrm{5}/\mathrm{2}} }{\mathrm{5}/\mathrm{2}}−\frac{{t}^{\mathrm{3}/\mathrm{2}} }{\mathrm{3}/\mathrm{2}}\right)=\frac{\mathrm{1}}{\mathrm{10}}{t}^{\mathrm{5}/\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{6}}{t}^{\mathrm{3}/\mathrm{2}} +{C} \\ $$

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