Question Number 198377 by sonukgindia last updated on 18/Oct/23
Answered by witcher3 last updated on 19/Oct/23
![=∫_0 ^∞ xe^(−4x^2 ) .xe^((−9)/x^2 ) dx IBP =[−(e^(−4x^2 ) /8)xe^(−(9/x^2 )) ]_0 ^∞ +(1/8)∫_0 ^∞ e^(−4x^2 ) (e^(−(9/x^2 )) +((18)/x^2 )e^(−(9/x^2 )) )dx I=(1/8)∫_0 ^∞ (1+((18)/x^2 ))e^(−4x^2 −(9/x^2 )) dx (1+((18)/x^2 ))=((13)/4)(2+(3/x^2 ))−((11)/4)(2−(3/x^2 )) I=((13)/(32))∫_0 ^∞ (2+(3/x^2 ))e^(−(2x−(3/x))^2 −12) −((11)/(32))∫_0 ^∞ (2−(3/x^2 ))e^(−(2x+(3/x))+12) =((13)/(32e^(12) ))(∫_0 ^(√(3/2)) (2+(3/x^2 ))e^(−(2x−(3/x))^2 ) +∫_(√(3/2)) ^∞ (2+(3/x^2 ))e^(−(2x−(3/x))^2 ) dx) u=2x−(3/x) =((13)/(32e^(12) ))∫_(−∞) ^∞ e^(−t^2 ) =((13)/(16e^(12) ))∫_0 ^∞ e^(−t^2 ) dt=((13(√π))/(32e^(12) )) ∫_0 ^∞ (2−(3/x^2 ))e^(−(2x+(3/x))^2 +12) =lim_(x→∞) ∫_(1/x) ^x (2−(3/t^2 ))e^(−(2t+(3/t))^2 +12) =lim_(x→∞) ∣∫_((2/x)+3x) ^(2x+(3/x)) e^(−t^2 +12) dt∣≤lim_(x→∞) ∫_(2x) ^(3x) e^(−t) dt→0 ∫_0 ^∞ x^2 e^(−(4x^2 +(9/x^2 ))) dx=((13(√π))/(32e^(12) ))](https://www.tinkutara.com/question/Q198430.png)
$$=\int_{\mathrm{0}} ^{\infty} \mathrm{xe}^{−\mathrm{4x}^{\mathrm{2}} } .\mathrm{xe}^{\frac{−\mathrm{9}}{\mathrm{x}^{\mathrm{2}} }} \mathrm{dx}\:\:\:\mathrm{IBP} \\ $$$$=\left[−\frac{\mathrm{e}^{−\mathrm{4x}^{\mathrm{2}} } }{\mathrm{8}}\mathrm{xe}^{−\frac{\mathrm{9}}{\mathrm{x}^{\mathrm{2}} }} \right]_{\mathrm{0}} ^{\infty} +\frac{\mathrm{1}}{\mathrm{8}}\int_{\mathrm{0}} ^{\infty} \mathrm{e}^{−\mathrm{4x}^{\mathrm{2}} } \left(\mathrm{e}^{−\frac{\mathrm{9}}{\mathrm{x}^{\mathrm{2}} }} +\frac{\mathrm{18}}{\mathrm{x}^{\mathrm{2}} }\mathrm{e}^{−\frac{\mathrm{9}}{\mathrm{x}^{\mathrm{2}} }} \right)\mathrm{dx} \\ $$$$\mathrm{I}=\frac{\mathrm{1}}{\mathrm{8}}\int_{\mathrm{0}} ^{\infty} \left(\mathrm{1}+\frac{\mathrm{18}}{\mathrm{x}^{\mathrm{2}} }\right)\mathrm{e}^{−\mathrm{4x}^{\mathrm{2}} −\frac{\mathrm{9}}{\mathrm{x}^{\mathrm{2}} }} \mathrm{dx} \\ $$$$\left(\mathrm{1}+\frac{\mathrm{18}}{\mathrm{x}^{\mathrm{2}} }\right)=\frac{\mathrm{13}}{\mathrm{4}}\left(\mathrm{2}+\frac{\mathrm{3}}{\mathrm{x}^{\mathrm{2}} }\right)−\frac{\mathrm{11}}{\mathrm{4}}\left(\mathrm{2}−\frac{\mathrm{3}}{\mathrm{x}^{\mathrm{2}} }\right) \\ $$$$\mathrm{I}=\frac{\mathrm{13}}{\mathrm{32}}\int_{\mathrm{0}} ^{\infty} \left(\mathrm{2}+\frac{\mathrm{3}}{\mathrm{x}^{\mathrm{2}} }\right)\mathrm{e}^{−\left(\mathrm{2x}−\frac{\mathrm{3}}{\mathrm{x}}\right)^{\mathrm{2}} −\mathrm{12}} −\frac{\mathrm{11}}{\mathrm{32}}\int_{\mathrm{0}} ^{\infty} \left(\mathrm{2}−\frac{\mathrm{3}}{\mathrm{x}^{\mathrm{2}} }\right)\mathrm{e}^{−\left(\mathrm{2x}+\frac{\mathrm{3}}{\mathrm{x}}\right)+\mathrm{12}} \\ $$$$=\frac{\mathrm{13}}{\mathrm{32e}^{\mathrm{12}} }\left(\int_{\mathrm{0}} ^{\sqrt{\frac{\mathrm{3}}{\mathrm{2}}}} \left(\mathrm{2}+\frac{\mathrm{3}}{\mathrm{x}^{\mathrm{2}} }\right)\mathrm{e}^{−\left(\mathrm{2x}−\frac{\mathrm{3}}{\mathrm{x}}\right)^{\mathrm{2}} } +\int_{\sqrt{\frac{\mathrm{3}}{\mathrm{2}}}} ^{\infty} \left(\mathrm{2}+\frac{\mathrm{3}}{\mathrm{x}^{\mathrm{2}} }\right)\mathrm{e}^{−\left(\mathrm{2x}−\frac{\mathrm{3}}{\mathrm{x}}\right)^{\mathrm{2}} } \mathrm{dx}\right) \\ $$$$\mathrm{u}=\mathrm{2x}−\frac{\mathrm{3}}{\mathrm{x}} \\ $$$$=\frac{\mathrm{13}}{\mathrm{32e}^{\mathrm{12}} }\int_{−\infty} ^{\infty} \mathrm{e}^{−\mathrm{t}^{\mathrm{2}} } =\frac{\mathrm{13}}{\mathrm{16e}^{\mathrm{12}} }\int_{\mathrm{0}} ^{\infty} \mathrm{e}^{−\mathrm{t}^{\mathrm{2}} } \mathrm{dt}=\frac{\mathrm{13}\sqrt{\pi}}{\mathrm{32e}^{\mathrm{12}} } \\ $$$$\int_{\mathrm{0}} ^{\infty} \left(\mathrm{2}−\frac{\mathrm{3}}{\mathrm{x}^{\mathrm{2}} }\right)\mathrm{e}^{−\left(\mathrm{2x}+\frac{\mathrm{3}}{\mathrm{x}}\right)^{\mathrm{2}} +\mathrm{12}} \\ $$$$=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\int_{\frac{\mathrm{1}}{\mathrm{x}}} ^{\mathrm{x}} \left(\mathrm{2}−\frac{\mathrm{3}}{\mathrm{t}^{\mathrm{2}} }\right)\mathrm{e}^{−\left(\mathrm{2t}+\frac{\mathrm{3}}{\mathrm{t}}\right)^{\mathrm{2}} +\mathrm{12}} \\ $$$$=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\mid\int_{\frac{\mathrm{2}}{\mathrm{x}}+\mathrm{3x}} ^{\mathrm{2x}+\frac{\mathrm{3}}{\mathrm{x}}} \mathrm{e}^{−\mathrm{t}^{\mathrm{2}} +\mathrm{12}} \mathrm{dt}\mid\leqslant\underset{{x}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{2x}} ^{\mathrm{3x}} \mathrm{e}^{−\mathrm{t}} \mathrm{dt}\rightarrow\mathrm{0} \\ $$$$\int_{\mathrm{0}} ^{\infty} \mathrm{x}^{\mathrm{2}} \mathrm{e}^{−\left(\mathrm{4x}^{\mathrm{2}} +\frac{\mathrm{9}}{\mathrm{x}^{\mathrm{2}} }\right)} \mathrm{dx}=\frac{\mathrm{13}\sqrt{\pi}}{\mathrm{32e}^{\mathrm{12}} } \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$