INTEGRATION BY SUBSTITUTION Meaning and
Definition
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Integration by substitution, also referred to as the method of change of variables, is a powerful technique used in calculus to simplify the integration of complicated functions. It is particularly effective when encountering integrals that involve composite functions or functions with multiple variables.
In this method, a substitution is made by rewriting the integral in terms of a new variable. This chosen substitution should transform the integrand into a simpler form that is easier to integrate. By differentiating the new variable and substituting it back into the integral, the expression can be simplified, facilitating the evaluation process.
The first step in integration by substitution is to identify a suitable substitution that simplifies the integral. This often involves selecting a substitution such that the differential of the new variable appears in the integrand. The next step is to substitute the new variable into the integral and rewrite it in terms of this variable. The third step entails differentiating the new variable and expressing it in terms of the original variable. Finally, the original integral is transformed into a new integral in terms of the new variable, thus making it easier to evaluate.
Integration by substitution is often used in cases where the integrand involves trigonometric functions, logarithmic functions, or algebraic functions. It allows the simplification of complex integrals and helps find antiderivatives that may be difficult or impossible to evaluate using other techniques. Integration by substitution not only simplifies the process of integration but also provides a deeper understanding of the relationship between variables, allowing for more efficient problem-solving in calculus.
