Integration by parts

Integration by parts handles a product of two different kinds of function, such as x sin x or x eˣ. It comes from the product rule, run backwards:

∫(first × second) dx = first × ∫second – ∫(derivative of first × ∫second) dx

Choosing the first function matters. Use the order ILATE, and take whichever comes earlier as first:

• I: inverse trig (tan⁻¹x)
• L: log (log x)
• A: algebraic (x, x²)
• T: trig (sin x)
• E: exponential (eˣ)

The idea: the first function should get simpler when differentiated (x becomes 1), and the second should be easy to integrate.

Example: ∫x sin x dx. First x, second sin x. = x(–cos x) – ∫1 × (–cos x) dx = –x cos x + sin x + C.

Only one function? Use 1 as the second function. ∫log x dx = ∫log x × 1 dx = x log x – ∫(1/x) × x dx = x log x – x + C. The same trick works for ∫tan⁻¹x dx.

A shortcut: if the integrand is eˣ[f(x) + f′(x)], the answer is simply eˣ f(x) + C. Example: ∫eˣ(1/x – 1/x²) dx: here f = 1/x and f′ = –1/x², so the answer is eˣ/x + C.

Three more standard results, found by parts:

• ∫√(a² – x²) dx = (x/2)√(a² – x²) + (a²/2) sin⁻¹(x/a) + C
• ∫√(x² + a²) dx = (x/2)√(x² + a²) + (a²/2) log|x + √(x² + a²)| + C
• ∫√(x² – a²) dx = (x/2)√(x² – a²) – (a²/2) log|x + √(x² – a²)| + C