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Let-f-x-0-x-e-t-2-dt-Prove-0-e-x-2-f-x-dx-e-pi-2-1-

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Question Number 135820 by Ar Brandon last updated on 16/Mar/21
Let f(x)=∫_0 ^x e^(−t^2 ) dt , Prove ∫_0 ^∞ e^(−x^2 +f(x)) dx=e^((√π)/2) −1.
Letf(x)=∫0xe−t2dt,Prove∫0∞e−x2+f(x)dx=eπ2−1.
Answered by mindispower last updated on 19/Mar/21
∫_0 ^∞ e^(−x^2 +f(x)) dx f′(x)=e^(−x^2 ) ⇔∫_0 ^∞ f′(x)e^(f(x)) dx=[e^(f(x)) ]_0 ^∞ =e^(∫_0 ^∞ e^(−t^2 ) dt) −1 =e^(√(π/4)) −1=e^((√π)/2) −1
∫0∞e−x2+f(x)dxf′(x)=e−x2⇔∫0∞f′(x)ef(x)dx=[ef(x)]0∞=e∫0∞e−t2dt−1=eπ4−1=eπ2−1

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