Question 1010978
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integrate {{{matrix(2,1,"",

(e^(x^2+2x))/((x+1)^2)))}}}
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1.  Multiply the numerator by  e  and divide the function by  e.  The integrand does not change,  it remains the same:


{{{matrix(2,1,"",

(e^(x^2+2x))/((x+1)^2)))}}} = {{{matrix(2,1,"",(1/e))}}}.{{{matrix(2,1,"",

(e^(x^2+2x+1))/((x+1)^2)))}}} = {{{matrix(2,1,"",(1/e))}}}.{{{matrix(2,1,"",

(e^((x+1))^2))/((x+1)^2)))}}}



2.  Replace the variable  y = x+1.  You will get


{{{matrix(2,1,"",(1/e))}}}.{{{matrix(2,1,"",

(e^((x+1)^2)*dx)/((x+1)^2)))}}} = {{{matrix(2,1,"",(1/e))}}}.{{{matrix(2,1,"",

(e^(y^2)*dy)/(y^2)))}}}



3.  Take the integral by parts:   u = {{{matrix(2,1,"",e^(y^2))}}},   v*dy = {{{(dy)/(y^2)}}},   v = {{{-1/y}}}. 


Do not forget this factor  {{{1/e}}}.


4.  In this way you will reduce the problem to the integral of the function   {{{matrix(2,1,"",e^(y^2))}}}. 


This integral is not an elementary function,  and it is entirely out of the school Calculus.  It is entirely out of the university Calculus,  even.