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Integrate x * e^x dx

Worked through step by step. More integrals.

Answer
xex−ex+C\boxed{x e^x - e^x + C}

Differentiating this result using the product rule yields ex+xex−ex=xexe^x + x e^x - e^x = x e^x, confirming the antiderivative is correct.

Problem: Evaluate the indefinite integral ∫xex dx\int x e^x \, dx.

Steps

  1. Integration by parts
    ∫u dv=uv−∫v du where u=x, dv=ex dx\int u \, dv = uv - \int v \, du \text{ where } u = x, \, dv = e^x \, dx
  2. Compute derivatives and integrals
    du=dx, v=exdu = dx, \, v = e^x
  3. Substitute into the integration by parts formula
    xex−∫ex dxx e^x - \int e^x \, dx
  4. Integrate the remaining term
    xex−ex+Cx e^x - e^x + C

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