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pivoxa15
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Homework Statement
What is the connection between the Indefinite integral and the Fundalmental theorem of calculus (1st part)?
The Attempt at a Solution
They are the same to me but the FT is more formal.
An indefinite integral is an expression in calculus that represents the antiderivative of a function. It is also known as a primitive function. It is used to find the original function when only its derivative is known.
The fundamental theorem of calculus is a theorem that connects the concepts of differentiation and integration. It states that if a function f(x) is continuous on an interval [a, b] and F(x) is an antiderivative of f(x) on [a, b], then the definite integral of f(x) on [a, b] is equal to F(b) - F(a). In other words, the definite integral of a function is equal to the difference between its antiderivative evaluated at the upper and lower bounds of the interval.
To solve an indefinite integral, you need to use integration techniques such as substitution, integration by parts, and partial fractions. First, identify the function and its limits. Then, use the appropriate integration technique to find the antiderivative of the function. Finally, add the constant of integration to get the final answer.
The main difference between indefinite and definite integrals is that indefinite integrals have no specified limits, while definite integrals have upper and lower bounds. Indefinite integrals represent a family of functions, while definite integrals give a specific numerical value. Additionally, definite integrals can be used to calculate area under a curve or find the net change in a function, while indefinite integrals are used to find the original function.
The fundamental theorem of calculus is important because it provides a connection between the two fundamental concepts of calculus - differentiation and integration. This theorem allows us to calculate definite integrals without using Riemann sums, making it an efficient and powerful tool in solving mathematical problems. It also helps us to better understand the relationship between a function and its antiderivative.