Integration By Parts

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Using integration by parts, ∫x·eˣ dx equals

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1. **Rule**: $\int u \, dv = uv - \int v \, du$.
2. **Choice of variables**: Use LIATE rule.
   - Let $u = x$ (algebraic) $\implies du = dx$
   - Let $dv = e^x \, dx$ (exponential) $\implies v = e^x$
3. **Substitution**:
   - $\int x \, e^x \, dx = (x)(e^x) - \int e^x \, dx$
   - $= x \, e^x - e^x + C$
4. **Simplification**: $e^x(x - 1) + C$.

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Integration By Parts

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