Integration by Parts

Integrating products of functions using the IBP formula

Golden Rule: Choose u as the factor that simplifies when differentiated. Follow LIATE: Logarithm, Inverse trig, Algebraic, Trig, Exponential.

• Step 1: Identify u and dv — Label the factor highest on the LIATE list as u, and everything else (including dx) as dv. If there is only one factor, set u = that factor and dv = dx.
• Step 2: Compute du and v — Differentiate your chosen u to find du. Integrate dv to find v — do not add a constant here. Keep the algebra clean before substituting.
• Step 3: Apply the formula — Write out the IBP formula and substitute your four pieces. Simplify the remaining integral ∫v du as much as possible before continuing.
• Step 4: Evaluate ∫v du — The new integral should be simpler than the original. If it is not, check your u/dv choice. If you land back at the original integral (e.g., ∫eˣsin x), move it to one side and solve algebraically.
• Step 5: Add the constant of integration — The full answer is uv − ∫v du + C. Do not forget + C on indefinite integrals.