Math Problem Statement

∫∞ 1 𝑥𝑒−𝑥𝑑𝑥

Solution

The given integral is:

1xexdx\int_1^\infty x e^{-x} \, dx

To solve this, we can use integration by parts.

Let:

  • u=xu = x, so du=dxdu = dx
  • dv=exdxdv = e^{-x} dx, so v=exv = -e^{-x}

Now applying the integration by parts formula:

udv=uvvdu\int u \, dv = uv - \int v \, du

Substitute the values:

1xexdx=[xex]1+1exdx\int_1^\infty x e^{-x} \, dx = \left[ -x e^{-x} \right]_1^\infty + \int_1^\infty e^{-x} \, dx

Step 1: Evaluate [xex]1\left[ -x e^{-x} \right]_1^\infty

At x=x = \infty: xex0(because ex0 faster than x)-x e^{-x} \to 0 \quad (\text{because } e^{-x} \to 0 \text{ faster than } x \to \infty)

At x=1x = 1: xex=e1-x e^{-x} = -e^{-1}

Thus: [xex]1=0(e1)=e1\left[ -x e^{-x} \right]_1^\infty = 0 - (-e^{-1}) = e^{-1}

Step 2: Solve the remaining integral 1exdx\int_1^\infty e^{-x} \, dx

1exdx=[ex]1\int_1^\infty e^{-x} \, dx = \left[ -e^{-x} \right]_1^\infty

At x=x = \infty: ex=0-e^{-x} = 0

At x=1x = 1: ex=e1-e^{-x} = -e^{-1}

Thus: 1exdx=0(e1)=e1\int_1^\infty e^{-x} \, dx = 0 - (-e^{-1}) = e^{-1}

Final Answer:

Now add the two results together: e1+e1=2e1e^{-1} + e^{-1} = 2e^{-1}

Thus, the value of the integral is:

1xexdx=2e\int_1^\infty x e^{-x} \, dx = \frac{2}{e}


Would you like further clarification on any of the steps?

Here are some related questions:

  1. What are other examples of integrals solved by integration by parts?
  2. How do we handle improper integrals like this one in general?
  3. Can we apply different techniques to evaluate integrals involving exe^{-x}?
  4. What happens if we change the limits of integration?
  5. How does the convergence of exe^{-x} affect integrals on an infinite interval?

Tip: Always check the behavior of terms at infinity when evaluating improper integrals.

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Math Problem Analysis

Mathematical Concepts

Calculus
Improper Integrals
Integration by Parts
Exponential Decay

Formulas

∫ u dv = uv - ∫ v du (Integration by Parts Formula)
∫_1^∞ e^(-x) dx = [ -e^(-x) ]_1^∞

Theorems

Integration by Parts
Behavior of Exponential Functions at Infinity

Suitable Grade Level

Undergraduate Calculus