What is answer of that integral? I gave a link here is the question ; if you help me i am very grateful thank you very much

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Answer 1 · 2018-05-01T18:37:34

#=(ln(5^5/3^3) - 2) sqrtpi #

Needed:

#int \ ln(y + 4) \ dy#

#= int \ ln z \ d(z-4) = int \ ln z \ d(z) qquad qquad " Now ready for IBP "#

#= z ln z - int \ d(ln z) z = #z ln z - z + C = (y+4) ln(y + 4) - y + C

The integral:

#int_(-oo)^(oo) int_(-1)^1 (xy + ln(y + 4))/e^(x^2) \ dy \ dx#

#= int_(-oo)^(oo)( (1/2 xy^2 + (y+4) ln(y + 4) - y)/e^(x^2) )_(-1)^1 \ dx#

#= int_(-oo)^(oo)( (1/2 x + 5 ln(5) - 1)/e^(x^2) - (1/2 x + (3) ln(3) +1)/e^(x^2) )\ dx#

#=(ln(5^5/3^3) - 2) color(red)(int_(-oo)^(oo) e^(-x^2) \ dx)#

#=(ln(5^5/3^3) - 2) color(red)(sqrtpi)#

[Red text is a standard result]