Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus is used to show the relationship that differentiation and integration operations in Mathematics are inverse of each other. The fundamental theorem of calculus has a rich history. However, the most important names associated with the theorem are Isaac Newton and Gottfried Leibniz. The notations used today are given by Gottfried Leibniz. We will discuss about the Second Fundamental Theorem of Calculus proof and Second Fundamental Theorem Of Calculus examples for better understanding and clarity of the topic.

History of Gottfried Leibniz

Gottfried Leibnitz

• Name: Gottfried Leibniz

• Born: 1 July 1646

• Died: 14 November 1716

• Field: Mathematics

• Nationality: German

Statement of Second Fundamental Theorem of Calculus

According to the Second Fundamental Theorem of Calculus, the differentiation of an antiderivative function results in original functions.

Mathematically, consider a function $f(x)$ defined over limits

$\dfrac{d}{d x}\left[\int_{a}^{x} f(t) \cdot d t\right]=f(x)$

Second Fundamental Theorem of Calculus Proof

Proof of the Second Fundamental Theorem

The proof includes three steps.

• Integrate the given function $f(t)$.

• Apply the upper and lower bounds of integration.

• Differentiate the obtained expression to finally get the initial function back.

$\dfrac{d}{d x}\left[\int_{a}^{x} f(t) \cdot d t\right]=\dfrac{d}{d x}[F(t)]_{a}^{x} \\$

$\Rightarrow \dfrac{d}{d x}[F(x)-F(a)]=f(x)$

The value of integration of $f(t)$ is $F(t)$.

Now, apply the upper bound limit of $x$ and the lower bound limit of $a$ for the function $F(x)$.

We will get,

$\Rightarrow F(x)-F(a)$ .

The derivation of $F(x)$ is equal to $f(x)$, which is the original function.

Hence the proof of the Second Fundamental Theorem of Calculus.

Limitations of the Second Fundamental Theorem of Calculus

• The function being continuous is a vital condition. The theorem is not applicable in the case of non-continuous functions.

• The second Fundamental Theorem of Calculus doesn't tell anything about antiderivatives as infinite antiderivatives are corresponding to different arbitrary constants.

Applications of the Second Fundamental Theorem of Calculus

• The second Fundamental Theorem of Calculus is used to relate the fundamental elements of calculus, i.e., integration and differentiation.

• It is used for some complex differentiation which will not be possible without using this theorem.

Second Fundamental Theorem of Calculus Examples

1. Find the differentiation of the anti-derivative of the function $\dfrac{1}{x}$ across the limits $x$ and $5$.

Ans: The given function is $f(x)=\dfrac{1}{x}$.

Let us find the antiderivative of this function across the limits from $x$ and $5$

$\int_{5}^{x} \dfrac{1}{x} \cdot d x=[\log x]_{5}^{x}$

$\Rightarrow \log x-\log 5$

Now, differentiating the obtained expression.

$\dfrac{d}{d x} \cdot(\log x-\log 5)=\dfrac{1}{x}-0$

$\Rightarrow \dfrac{1}{x}$

Using the second fundamental theorem of calculus,

$\Rightarrow \dfrac{d}{d x} \int_{5}^{x} \dfrac{1}{x}=\dfrac{1}{x}$

Therefore, the differentiation of the anti-derivative of the function $\dfrac{1}{x}$ is $\dfrac{1}{x}$.

2. Prove that the differentiation of the anti-derivative of the function cosx will give the same function.
Ans: The given function is $f(x)=\cos x$

The integration of $\cos x$ is equal to the function $\sin x$.

$\int \cos x \cdot d x=\sin x+C$

Now, differentiating $\sin x+C$,

$\Rightarrow \dfrac{d}{d x}(\sin x+C)=\cos x$

So,

$\Rightarrow \dfrac{d}{d x} \int \cos x d x=\cos x$

3. Find the value of the integral \[\int_{-1}^3\left(x^2-3\right) d x\] using the fundamental theorem of calculus.

Ans: Using FTC 2, \[\int_{a}^b f(x) d x=F(b)-F(a)\] where \[F(x)=\int f(x) d x\].

So, first, we will evaluate the indefinite integral \[\int\left(x^2-3\right) d x\].

\[\int\left(x^2-3\right) d x=\int\left(x^2\right) d x-\int 3 d x \]

\[=\dfrac{x^3}{3}-3 x+C \quad\left(\because \int x^n d x=\dfrac{x^{n+1}}{(n+1)}+C\right) \]

\[=F(x)\]

BY FTC 2,

\[\int_{-1}^3\left(x^2-3\right) d x=\dfrac{x^3}{3}-3 x+C \]

\[=\left[\dfrac{3^3}{3}-3(3)\right]-\left[\dfrac{(-1)^3}{3}-3(-1)\right] \]

\[=9-9+\dfrac{1}{3}-3 \]

\[=-\dfrac{8}{3}\]

Important Formulas to Remember

• For a continuous function $f(x)$, we have $\dfrac{d}{d x}\left[\int_{a}^{x} f(t) \cdot d t\right]=f(x)$.

Important Points to Remember

• Integration and differentiation are related to each other with the help of the Second Fundamental Theorem of Calculus.

• There are a total of three fundamental theorems of calculus.

Conclusion

In the article, we have discussed the detailed proof of the Second Fundamental Theorem of Calculus and its applications. The theorem is quite fundamental in its sense and hence named so. The novel thing about the theorem is the way it connects two different fundamental tools with a relation. This theorem is the backbone of calculus.