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1-x-2-3-1-x-dx-




Question Number 7482 by kishanjith last updated on 31/Aug/16
∫(((1+x)^(−2/3) )/((1+x)))dx
$$\int\frac{\left(\mathrm{1}+{x}\right)^{−\mathrm{2}/\mathrm{3}} }{\left(\mathrm{1}+{x}\right)}{dx} \\ $$
Answered by sandy_suhendra last updated on 31/Aug/16
=∫(1−x)^((−5)/3) dx  let   U=1−x   then   (dU/dx) = −1 ⇒ dx = −dU  =∫ U^((−5)/3)  (−dU)  =−∫ U^((−5)/3)  dU  = (3/2) U^((−2)/3) + c  =(3/2) (1−x)^((−2)/3) + c
$$=\int\left(\mathrm{1}−{x}\right)^{\frac{−\mathrm{5}}{\mathrm{3}}} {dx} \\ $$$${let}\:\:\:{U}=\mathrm{1}−{x}\: \\ $$$${then}\:\:\:\frac{{dU}}{{dx}}\:=\:−\mathrm{1}\:\Rightarrow\:{dx}\:=\:−{dU} \\ $$$$=\int\:{U}^{\frac{−\mathrm{5}}{\mathrm{3}}} \:\left(−{dU}\right) \\ $$$$=−\int\:{U}^{\frac{−\mathrm{5}}{\mathrm{3}}} \:{dU} \\ $$$$=\:\frac{\mathrm{3}}{\mathrm{2}}\:{U}^{\frac{−\mathrm{2}}{\mathrm{3}}} +\:{c} \\ $$$$=\frac{\mathrm{3}}{\mathrm{2}}\:\left(\mathrm{1}−{x}\right)^{\frac{−\mathrm{2}}{\mathrm{3}}} +\:{c} \\ $$

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