Faktorhelyettesítés

Faktorhelyettesítés, also known as substitution or change of variables, is a fundamental technique used in mathematics, particularly in calculus and algebra. It involves replacing a part of an expression or equation with a new variable. This substitution simplifies the original expression, making it easier to solve or manipulate. The process typically involves defining a new variable, say 'u', that is equal to a complex part of the original expression, like a term within a function or an integrand. Then, the original expression is rewritten entirely in terms of 'u'.

After the substitution, the problem is solved using the new variable 'u'. Once a solution is found in terms of 'u', the original expression is substituted back in place of 'u' to obtain the final answer in terms of the original variables. This method is widely used for solving integrals, differentiating complex functions, and simplifying algebraic equations. For instance, in integration, if an integral contains a composite function, substituting the inner function with a new variable can transform a difficult integral into a standard form that is readily solvable. The key to successful faktorhelyettesítés lies in correctly identifying the part of the expression to substitute and accurately transforming all other parts, including differentials if applicable, into the new variable.