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zorro
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If we have a definite integral given by f(x) = ∫sintdt with limits from 0 to x, can we integrate the function directly by substituting t as x because the variable involved in the integrand is a dummy variable?
A definite integral is a mathematical concept used in calculus to find the area under a curve between two specific points on a graph. It is represented by the symbol ∫ and is used to find the total value of a continuous function over a given interval.
A dummy variable is a placeholder variable that is used in place of the variable in the original function when evaluating a definite integral. It is often denoted by a letter such as x or t and is used to represent the independent variable in the function.
To evaluate a definite integral with a dummy variable, you first substitute the dummy variable with the limits of integration. Then, you integrate the function with respect to the dummy variable and evaluate the resulting expression at the upper limit, subtracting the value at the lower limit.
The purpose of using a dummy variable in a definite integral is to allow for a more general and flexible approach to evaluating integrals. It also helps to avoid confusion when there are multiple variables involved in a function.
No, substitution is an essential step in evaluating a definite integral with a dummy variable. It allows for the proper integration of the function and ensures that the result is accurate. Attempting to solve the integral without substitution may lead to incorrect results.