What is integration by parts, with examples?
Quick answer
Integration by parts is used to integrate a product of two functions. The formula is ∫u·dv = u·v − ∫v·du. Choose u using the ILATE rule (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential) — whichever comes first becomes u.
Example 1: ∫x·eˣ dx. Let u = x, dv = eˣdx, so du = dx, v = eˣ. Then ∫x·eˣ dx = x·eˣ − ∫eˣ dx = eˣ(x − 1) + C.
Example 2: ∫x·ln x dx. Let u = ln x, dv = x dx, so du = dx/x, v = x²/2. Then the integral = (x²/2)ln x − ∫(x/2) dx = (x²/2)ln x − x²/4 + C.
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