Let \(f(x) = \int xe^{x}\,dx\). Then \(f(x)\) decreases in the interval
\((-\infty, 0)\)
Integrating by parts, \(\int xe^{x}\,dx = xe^{x} - e^{x} + C = e^{x}(x-1)+C\), so \(f(x) = e^{x}(x-1)+C\). Differentiating, \(f'(x) = e^{x}(x-1)+e^{x} = xe^{x}\). Since \(e^{x} > 0\) for all \(x\), \(f'(x) < 0\) exactly when \(x < 0\). Hence \(f(x)\) decreases on \((-\infty, 0)\).
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