To evaluate this integral, we first need to simplify the exponential expression inside it, by dividing each term in the numerator by the denominator `e^x` .
The first term would become `(xe^x)/e^x = x` because the exponent `e^x` cancels.
The second term would become `e^(2x-1)/e^x = e^(2x-1-x) = e^(x-1)` (Here, the rule of exponent is applied: to divide the powers of the same base, subtract exponents. This could be further rewritten as `e^(x-1) = e^x/e`...
To evaluate this integral, we first need to simplify the exponential expression inside it, by dividing each term in the numerator by the denominator `e^x` .
The first term would become `(xe^x)/e^x = x` because the exponent `e^x` cancels.
The second term would become `e^(2x-1)/e^x = e^(2x-1-x) = e^(x-1)` (Here, the rule of exponent is applied: to divide the powers of the same base, subtract exponents. This could be further rewritten as `e^(x-1) = e^x/e` , 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
`x - e^x/e` , which is a difference of a power function and an exponential function. The integral of a difference is a difference of integrals, so
`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:
`int xdx = x^2/2` (up to a constant) and `int e^x dx = e^x` (up to a constant.)
The final result is therefore `x^2/2 - 1/e*e^x + C = x^2/2 - e^(x-1) + C` , where C is a constant.
The integral in question equals `x^2/2 - e^(x-1) + C` .
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