Understanding the Fundamentals: Evaluating the Integral with Same Limits of Integration

Integral from a to a of f(x) with respect to x

The integral from a to a of f(x) with respect to x is defined as:

∫ [a to a] f(x) dx

To evaluate this integral, we can use the fundamental theorem of calculus, which states that if F(x) is an antiderivative of f(x), then:

∫ [a to b] f(x) dx = F(b) – F(a)

In this case, the limits of integration are the same, a to a

The integral from a to a of f(x) with respect to x is defined as:

∫ [a to a] f(x) dx

To evaluate this integral, we can use the fundamental theorem of calculus, which states that if F(x) is an antiderivative of f(x), then:

∫ [a to b] f(x) dx = F(b) – F(a)

In this case, the limits of integration are the same, a to a. Since a is a constant, any constant value integrated with respect to x will result in a multiple of x. Therefore, the integral from a to a of f(x) will be:

∫ [a to a] f(x) dx = F(a) – F(a)

Notice that the two terms F(a) cancel each other out because they are subtracted. Therefore, the final result is zero:

∫ [a to a] f(x) dx = 0

In simpler terms, if the limits of integration are the same, the integral of any function from a to a will be zero.

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