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Question-97059




Question Number 97059 by mhmd last updated on 06/Jun/20
Commented by PRITHWISH SEN 2 last updated on 06/Jun/20
let      ((x^2 +1)/(sin^(−1) x−1)) = v(x)  then v^′ (x)=((2xsin^(−1) x.(√(1−x^2 ))−2x(√(1−x^2 ))−x^2 −1)/( (√(1−x^2 )) (sin^(−1) x−1)^2 ))  ∫{e^x v(x)+e^x v′(x) }dx  using by parts  e^x v(x)−∫e^x v′(x)+∫e^x v^′ (x) +C  =e^x v(x)+C=e^x .((x^2 +1)/(sin^(−1) x−1)) +C  please check
$$\mathrm{let}\: \\ $$$$\:\:\:\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}{\mathrm{sin}^{−\mathrm{1}} \mathrm{x}−\mathrm{1}}\:=\:\mathrm{v}\left(\mathrm{x}\right) \\ $$$$\mathrm{then}\:\mathrm{v}^{'} \left(\mathrm{x}\right)=\frac{\mathrm{2xsin}^{−\mathrm{1}} \mathrm{x}.\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }−\mathrm{2x}\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }−\mathrm{x}^{\mathrm{2}} −\mathrm{1}}{\:\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\:\left(\mathrm{sin}^{−\mathrm{1}} \mathrm{x}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\int\left\{\mathrm{e}^{\mathrm{x}} \mathrm{v}\left(\mathrm{x}\right)+\mathrm{e}^{\mathrm{x}} \mathrm{v}'\left(\mathrm{x}\right)\:\right\}\mathrm{dx} \\ $$$$\mathrm{using}\:\mathrm{by}\:\mathrm{parts} \\ $$$$\mathrm{e}^{\mathrm{x}} \mathrm{v}\left(\mathrm{x}\right)−\int\mathrm{e}^{\mathrm{x}} \mathrm{v}'\left(\mathrm{x}\right)+\int\mathrm{e}^{\mathrm{x}} \mathrm{v}^{'} \left(\mathrm{x}\right)\:+\mathrm{C} \\ $$$$=\mathrm{e}^{\mathrm{x}} \mathrm{v}\left(\mathrm{x}\right)+\mathrm{C}=\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} .\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{1}}{\mathrm{sin}^{−\mathrm{1}} \boldsymbol{\mathrm{x}}−\mathrm{1}}\:+\mathrm{C}\:\:\boldsymbol{\mathrm{please}}\:\boldsymbol{\mathrm{check}} \\ $$

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