- #1
Grimertop90
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I feel really dumb for not remembering this... I'm drawing a total blank as to the antiderivative of xe-x or how to find it. Do I need to use a u substitution?
The integral of xe^-x is -(x+1)e^-x + C, where C is the constant of integration.
To solve for the integral of xe^-x, we can use the integration by parts method. We let u = x and dv = e^-x, and then apply the formula ∫u dv = uv - ∫v du.
The constant of integration is necessary because when we take the derivative of -(x+1)e^-x, we get xe^-x. However, when we take the derivative of -(x+1)e^-x + C, the C term becomes 0 and we are left with xe^-x, which is the original function we started with. Therefore, the constant of integration accounts for all possible solutions to the integral.
No, the integral of xe^-x cannot be simplified further. It is already in its simplest form, -(x+1)e^-x + C.
Yes, there are many real world applications of the integral of xe^-x. One example is in physics, where the integral can be used to solve for the displacement of an object given its acceleration and initial velocity. It is also commonly used in economics and finance to calculate the present value of future cash flows.