How in indefinite integral we can do the substitution without bothering about the variable or function we are substituting the variable with? For example, it is very common to make trigonometric substitutions for evaluating indefinite integrals like we put x = sint or cosect or cost etc. But how can we do that when we know that the original variable x can assume any real values but the function sint, cost by definition can only asse values in [-1,1] . Now It is okay to make the substitutions x in terms of tan or cot as they can assume all real values. But how is it justified for other trig functions? Also why in indefinite integrals we generally ignore the modulus or domain etc. As in Paul's online notes http://tutorial.math.lamar.edu/Classes/CalcII/TrigSubstitutions.aspx While solving and only take care in definite integral?
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