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Second Fundamental Theorem of Calculus
The second Fundamental Theorem of Calculus gives a relationship between the two processes of differentiation and integration, which are inverse processes of each other. For a function f(x) the differentiation of the antiderivative of the function results back in the original function.
Let us learn more about the statement, proof of the second fundamental theorem of calculus, with the help of examples, FAQs.
What Is Second Fundamental Theorem of Calculus?
The Second Fundamental Theorem of Calculus relates the two operations of differentiation and integration and shows that these two operations are inverse of each other. This theorem connects the differentiation of a function which is the gradient of a curve with the integration of the function which is the area enclosed under the curve. The second fundamental theorem states that the differentiation of the antiderivative of a function gives back the original function.
Statement: If a function f is continuous over the interval [a, b] and differentiable across the interval (a, b) then the differentiation of the antiderivative of the function gives back the function f.
\(\dfrac{d}{dx}\int^x_af(x).dx = f(x)\)
Proof of Second Fundamental Theorem of Calculus
The proof includes three simple steps. First, the given function f(t) is integrated, secondly, the upper and lower bounds of integration are applied, and finally, the differentiation of this expression gives back the initial function.
\(\begin{align}\dfrac{d}{dx}\left[\int ^x_af(t).dt\right]&=\dfrac{d}{dx}[F(t)]^x_a \\&=\dfrac{d}{dx}[F(x)  F(a)] \\&= F'(x) = f(x)\end{align}\)
The integration of f(t) is equal to F(t). Further, the upper bound limit of x and the lower bound limit of a is applied for the function F(x), to obtain F(x)  F(a). The derivation of F(x) is equal to F'(x), which is equal to f(x), the original function.
Related Topics
The following topics are helpful for a better understanding of the second fundamental theorem of calculus.
Examples on Second Fundamental Theorem of Calculus

Example 1: Find the differentiation of the antiderivative of the function 1/x across the limits x and 5, by using the concepts from the second fundamental theorem of calculus..
Solution:
The given function is f(x) = 1/x
Let us find the antiderivative or the integration of this function across the limits from x and 5.
\(\int^x_5 \dfrac{1}{x}.dx =[logx]^x_5 = logx  log5\)
Further let us find the differentiation of the obtained expression.
\( \dfrac{d}{dx}.(logx  log5) = \dfrac{1}{x}  0 = \dfrac{1}{x} \)
Thus applying the second fundamental theorem of calculus, the above two processes of differentiation and antiderivative can be shown in a single step.
\( \dfrac{d}{dx}\int^x_5\dfrac{1}{x} = \dfrac{1}{x} \)
Therefore, the differentiation of the antiderivative of the function 1/x is 1/x.

Example 2: Prove that the differentiation of the antiderivative of the function Cosx, results in the same function.
Solution:
The given function is f(x) = Cosx.
The integration of cosx gives the function Sinx.
\(\int Cosx.dx = Sinx + C\).
Further differentiation of Sinx + C results in the function of Cosx.
\(\dfrac{d}{dx}.(Sinx + C).dx = Cosx\).
Thus by combining both the above operations through the application of the second fundamental theorem of calculus we have the following expression.
\( \dfrac{d}{dx}\int Cosx = Cosx \)
Therefore, we have successfully proved that the differentiation and the antiderivative of the function Cosx is the same function of Cosx.
FAQs on Second Fundamental Theorem of Calculus
What Is The Second Fundamental Theorem of Calculus?
The second fundamental theorem of calculus gives a holistic relationship between the two processes of integration and differentiation. It states that, if a function f is continuous over the interval [a, b] and differentiable across the interval (a, b) then the differentiation of the antiderivative of the function gives back the function f. This is expressed in the form of a mathematical expression as \(\dfrac{d}{dx}\int^x_af(x).dx = f(x)\).
What Is The Formula Of Second Fundamental Theorem of Calculus?
The formula for the second fundamental theorem of calculus is \(\dfrac{d}{dx}\int^x_af(x).dx = f(x)\). Here the differentiation of the antiderivative of the function f(x) is computed, which is equal to the initial function f(x).
What Is The Use Of Second Fundamental Theorem of Calculus?
The second fundamental theorem of calculus is useful to understand the relationship between the two important operations of differentiation and integration. Integration is also referred as antiderivative, and this theorem is useful to understand that integration and differentiation are the reverse processes of each other.
What Is The Difference Between First Fundamental Theorem Of Calculus And Second Fundamental Theorem Of Calculus?
The first fundamental theorem of calculus explains the antiderivative of a function with an upper bound, lower bound, and the second fundamental theorem of calculus gives a relationship between the process of differentiation and integration. The first fundamental theorem of calculus states that the integration or the antiderivative of a function can be taken with an upper bound and a lower bound. \( [\int^a_b f(x).dx] = F(b)  F(a) \). The second fundamental theorem of calculus states that the differentiation of the antiderivative of a function gives back the initial function. \(\dfrac{d}{dx}\int^x_af(x).dx = f(x)\)
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