Can you help me to solve the indefinite integral `int (xe^x - e^(2x-1))/e^x` ? I don't understand how solve it! Please, help me!
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` .