Integration by Parts Guide
Tabular integration for ∫ xᵐ(ax+b)ⁿ dx: build the table, multiply diagonally, alternate signs.
Example: ∫ x²·x³ dx = ∫x⁵ dx. Try m = 2, a = 1, b = 1, n = 2 for ∫ x²(x+1)² dx.
① Answer
② Steps
③ Why it works
Integration by parts comes from the product rule: ∫ u·dv = u·v − ∫ v·du. When one factor is a polynomial, the tabular method does all the rounds at once: differentiate the polynomial column until it hits zero, integrate the other column the same number of times, then multiply diagonally with alternating signs (+, −, +, −, …).