Integrating ∫xeˣ dx

Calculus 2 · Axiom Academy

Master the classic integration-by-parts technique by evaluating step-by-step. Evaluate the integral . The integrand is a product of two functions ( x and e^x ), which is the signal to reach for integration by parts : . Nice work! You evaluated with integration by parts. Here's what carries forward: LIATE rule: When choosing u , use the priority order Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Since x is algebraic and e^x is exponential, u = x . Strategic selection: Pick u to be the factor that simplifies when differentiated, and dv to be the factor that's easy to integrate. Apply the formula: With u , du , v , and dv identified, substitute into . Simplify the new integral: Integration by parts trades the original integral for a simpler one — here . Always verify: Differentiate the answer. , the original integrand. Integration by parts is essential whenever an integrand is a product of functions — polynomials with exponentials, logarithms, or trig — and it reappears throughout Calculus 2 and beyond.

This is the written version of the interactive lesson above. See the full Calculus 2 course.