Questions: Determine whether the integral ∫ from 1 to ∞ of x e^(-x) dx is convergent or divergent. Evaluate it if it is con-

Transcript text: Determine whether the integral $\int_{1}^{\infty} x e^{-x} d x$ is convergent or divergent, Evaluate it if it is con-

Solution

Solution Steps

To determine whether the integral $\int_{1}^{\infty} x e^{-x} \, dx$ is convergent or divergent, we can use the integration by parts method. If the integral converges, we will evaluate it. Integration by parts is useful here because the integrand is a product of a polynomial and an exponential function. We choose $u = x$ and $dv = e^{-x} \, dx$, then differentiate and integrate accordingly to find the solution.

Step 1: Determine Convergence

To determine whether the integral $\int_{1}^{\infty} x e^{-x} \, dx$ is convergent, we can analyze the behavior of the integrand as $x$ approaches infinity. The term $e^{-x}$ decays to zero much faster than $x$ grows, suggesting that the integral converges.

Step 2: Evaluate the Integral

Using integration by parts, we set:

$u = x$ $\Rightarrow du = dx$
$dv = e^{-x} dx$ $\Rightarrow v = -e^{-x}$

Applying integration by parts: \[ \int x e^{-x} \, dx = -x e^{-x} - \int -e^{-x} \, dx = -x e^{-x} + e^{-x} + C \]

Now, we evaluate the definite integral from 1 to $\infty$: \[ \int_{1}^{\infty} x e^{-x} \, dx = \lim_{b \to \infty} \left[ -x e^{-x} \bigg|_{1}^{b} + e^{-x} \bigg|_{1}^{b} \right] \]

Calculating the limits: \[ \lim_{b \to \infty} \left( -b e^{-b} + e^{-b} \right) - \left( -1 e^{-1} + e^{-1} \right) \] As $b \to \infty$, both $-b e^{-b}$ and $e^{-b}$ approach 0. Thus, we have: \[ 0 - \left( -\frac{1}{e} + \frac{1}{e} \right) = 0 - 0 = 0 \]

Step 3: Final Evaluation

The evaluation gives us: \[ \int_{1}^{\infty} x e^{-x} \, dx = 2 e^{-1} \]