To evaluate this integral, we first need to simplify the exponential expression inside it, by dividing each term in the numerator by the denominator 
.
The first term would become 
because the exponent cancels.
The second term would become 
(Here, the rule of exponent is applied: to divide the powers of the same base, subtract exponents. This could be further rewritten as 
, again, by applying the same rule of exponents. Remember that e is just a constant, which could be taken out of the integral.
So the expression under the integral, once simplified, becomes
, which is a difference of a power function and an exponential function. The integral of a difference is a difference of integrals, so
%7D%7B%5Cleft.%7Bd%7D%7Bx%7D%5Cright.%7D%3D%5Cint%7B%5Cleft(%7Bx%7D%5Cright)%7D%7B%5Cleft.%7Bd%7D%7Bx%7D%5Cright.%7D-%5Cint%7B%5Cleft(%5Cfrac%7B%7B%7Be%7D%7D%5E%7B%7Bx%7D%7D%7D%7B%7Be%7D%7D%5Cright)%7D%7B%5Cleft.%7Bd%7D%7Bx%7D%5Cright.%7D%3D%5Cint%7B%5Cleft(%7Bx%7D%5Cright)%7D%7B%5Cleft.%7Bd%7D%7Bx%7D%5Cright.%7D-%5Cfrac%7B%7B1%7D%7D%7B%7Be%7D%7D%5Cint%7B%5Cleft(%7B%7Be%7D%7D%5E%7B%7Bx%7D%7D%5Cright)%7D%7B%5Cleft.%7Bd%7D%7Bx%7D%5Cright.%7D "int(x-e^x/x) dx = int(x)dx - int(e^x/e)dx = int(x) dx- 1/e int(e^x)dx")
These integrals can now be evaluated:
(up to a constant) and 
(up to a constant.)
The final result is therefore 
, where C is a constant.
The integral in question equals 
.
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