2nd Fundamental Theorem of Calculus
d/dx the integral of f(t) from a to x = f(x)
The 2nd Fundamental Theorem of Calculus states that:
If f(x) is a continuous function on [a,b] and F(x) is an antiderivative of f(x) on [a,b], then the definite integral of f(x) from a to b is equal to F(b) – F(a), or in mathematical notation:
∫a^b f(x) dx = F(b) – F(a)
This theorem provides us with a way to evaluate definite integrals without having to resort to the limit definition of the integral. Instead, we can find an antiderivative of the integrand and use the formula above.
For example, let’s say we want to evaluate the definite integral of x^2 dx from 0 to 2. Using the 2nd Fundamental Theorem of Calculus, we can find an antiderivative of x^2, which is x^3/3. Then, we can substitute 2 into this antiderivative and subtract the result we get by substituting 0 into it:
∫0^2 x^2 dx = [x^3/3]2 – [x^3/3]0 = (2^3/3) – (0^3/3) = 8/3
Therefore, the definite integral of x^2 dx from 0 to 2 is equal to 8/3.
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