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!

Expert Answers
ishpiro eNotes educator| Certified Educator

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` .