Osittaisintegrointi

Osittaisintegrointi, known in English as integration by parts, is a method used in calculus to find integrals of products of functions. It is derived from the product rule for differentiation. The product rule states that the derivative of a product of two functions, u and v, is given by (uv)' = u'v + uv'. Integrating both sides of this equation with respect to x yields uv = ∫u'v dx + ∫uv' dx. Rearranging this equation, we get ∫uv' dx = uv - ∫u'v dx. This is the fundamental formula for integration by parts.

The practical application involves choosing one part of the integrand to be u and the other to be dv. The choice is crucial and often depends on which part simplifies when differentiated (becomes u') and which part is easily integrable (becomes v). Common heuristics, like the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential), suggest an order for choosing u to simplify the process.

Integration by parts is particularly useful for integrating products of different types of functions, such as a polynomial multiplied by an exponential function or a logarithmic function multiplied by a polynomial. It can also be used to integrate functions that do not initially appear as products, by considering them as a product with 1. For example, to integrate ln(x), one can treat it as ln(x) * 1, letting u = ln(x) and dv = 1 dx. This technique is a cornerstone of integral calculus, enabling the solution of many otherwise intractable integrals.