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Question-98594




Question Number 98594 by bemath last updated on 15/Jun/20
Answered by bobhans last updated on 15/Jun/20
∫ e^(x^2 /2)  (x^2 +2x+1) dx = ∫ e^(x^2 /2)  x^2  dx + ∫ 2x e^(x^2 /2)  dx + ∫ e^(x^2 /2)  dx  I_1  = ∫e^(x^2 /2) x^2  dx = ∫x e^(x^2 /2)  d((x^2 /2)) = xe^(x^2 /2)  −∫ e^(x^2 /2)  dx  I_2  = ∫ 2xe^(x^2 /2)  dx = 2∫ e^(x^2 /2)  d((x^2 /2)) = 2e^(x^2 /2)   I_3  = ∫ e^(x^2 /2)  dx   ∴I = I_1  + I_2  + I_3  = xe^(x^2 /2) + 2e^(x^2 /2)  = (x+2) e^(x^2 /2)  + C
$$\int\:\mathrm{e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}} \:\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{1}\right)\:\mathrm{dx}\:=\:\int\:\mathrm{e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}} \:\mathrm{x}^{\mathrm{2}} \:\mathrm{dx}\:+\:\int\:\mathrm{2x}\:\mathrm{e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}} \:\mathrm{dx}\:+\:\int\:\mathrm{e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}} \:\mathrm{dx} \\ $$$$\mathrm{I}_{\mathrm{1}} \:=\:\int\mathrm{e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}} \mathrm{x}^{\mathrm{2}} \:\mathrm{dx}\:=\:\int\mathrm{x}\:\mathrm{e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}} \:\mathrm{d}\left(\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}\right)\:=\:\mathrm{xe}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}} \:−\int\:\mathrm{e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}} \:\mathrm{dx} \\ $$$$\mathrm{I}_{\mathrm{2}} \:=\:\int\:\mathrm{2xe}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}} \:\mathrm{dx}\:=\:\mathrm{2}\int\:\mathrm{e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}} \:\mathrm{d}\left(\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}\right)\:=\:\mathrm{2e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}} \\ $$$$\mathrm{I}_{\mathrm{3}} \:=\:\int\:\mathrm{e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}} \:\mathrm{dx}\: \\ $$$$\therefore\mathrm{I}\:=\:\mathrm{I}_{\mathrm{1}} \:+\:\mathrm{I}_{\mathrm{2}} \:+\:\mathrm{I}_{\mathrm{3}} \:=\:\mathrm{xe}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}} +\:\mathrm{2e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}} \:=\:\left(\mathrm{x}+\mathrm{2}\right)\:\mathrm{e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}} \:+\:\mathrm{C} \\ $$

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