Evaluating ∫ x·eˣ dx

Real Analysis · Axiom Academy

Work through step by step using integration by parts. Evaluate the indefinite integral . The integrand is a product of two functions, so we will use the integration-by-parts technique. Nice work! You evaluated by integration by parts. Keep these points in mind: The formula: turns a hard product into a uv term minus an easier integral. Choose u wisely: pick the factor that simplifies when differentiated — here u = x becomes du = dx . Choose dv strategically: pick the factor that is easy to integrate — here e^x integrates to itself. LIATE rule: Logarithmic, Inverse-trig, Algebraic, Trigonometric, Exponential — Algebraic ( x ) comes before Exponential ( e^x ), so u = x . Don't forget +C : always include the constant of integration for an indefinite integral. Result: . The same idea extends to (apply parts twice) and .

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