Integration by Parts: ILATE Rule with Solved Examples

Substitution reverses the chain rule, but what reverses the product rule? When the integrand is a product of two different kinds of functions — like x·sin x or x·eˣ — neither function is the derivative of the other's inner part, so substitution fails. Integration by parts fills that gap: it trades the original product integral for a new one that is hopefully simpler.

This method matters because products of polynomials with exponentials, logarithms and trig functions are everywhere in science: the expected value of a probability distribution, the work done by a varying force, and solutions to differential equations all lead to such products. Choosing correctly which part to differentiate is a skill that pays off across the whole of calculus.

Key Concepts

• The formula: ∫ u dv = uv − ∫ v du. Choose u to differentiate and dv to integrate.
• ILATE rule: pick u by priority — Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential. The function appearing first becomes u.
• Why ILATE works: differentiating logarithms and inverse trig simplifies them dramatically, while integrating exponentials and trig leaves them almost unchanged.
• Repeated parts: products like x²eˣ may need the method applied two or more times.
• Cyclic integrals: for ∫ eˣ sin x dx, applying parts twice returns the original integral, which you then solve as an equation.

Formulas

• ∫ u dv = uv − ∫ v du
• ∫ x eˣ dx = eˣ(x − 1) + C
• ∫ x sin x dx = −x cos x + sin x + C
• ∫ ln x dx = x ln x − x + C (treat as ∫ 1 · ln x dx, with u = ln x)
• ∫ eˣ sin x dx = eˣ(sin x − cos x)/2 + C

Worked Example

Evaluate ∫ x sin x dx. Step 1: by ILATE, algebraic x comes before trigonometric sin x, so take u = x and dv = sin x dx. Then du = dx and v = −cos x. Step 2: apply the formula: ∫ x sin x dx = x(−cos x) − ∫ (−cos x) dx = −x cos x + ∫ cos x dx. Step 3: the remaining integral is standard: −x cos x + sin x + C. So ∫ x sin x dx = −x cos x + sin x + C. Differentiating the answer gives x sin x, which confirms it.

Common Misconceptions

• Choosing u backwards: taking u = sin x in the example above leads to ∫ x² cos x dx — harder, not easier. ILATE exists precisely to avoid this.
• Forgetting the second integral: the formula has a minus sign and a whole new integral; students often write uv and stop.
• Using parts when substitution fits: ∫ x·e^(x²) dx is a chain-rule reversal (substitute u = x²), not a parts problem.

Real-World Applications

Integration by parts derives the formula for the mean of a continuous probability distribution and appears in the moment-generating functions of statistics. In physics it evaluates work and centre-of-mass integrals with polynomial-times-trig forms. Engineers use it to solve the differential equations of vibrating systems and heat flow.

Integration by parts is the product rule in reverse: pick u with ILATE, apply uv minus the new integral, and simplify. When the new integral is easier than the old one, you are on the right track; if it gets harder, swap your choice.