Newton-Leibniz Formula

Differentiation and integration play a very important part in the mathematical studies of higher-order. And, in their very concept, they cover a very large basis for their varied applicability.  The method of Newton-Leibnitz is used to solve the definite integrals, but of higher difficulties, the ones in which the limit for the integral in itself is a  differential variable. To solve such kinds of problems, a different set of methods is to be used, and we get that type of method from this theorem. 

Fundamental Theorem of Calculus

The fundamental theorem of calculus is the theorem that relates the differentiation and integration of a function together with each other. These two operations which are countered by each other, are united by a constant, over which the computation of these values starts. 

The first fundamental theorem of calculus is applied to the concept of definite integrals to solve integration problems. 

The result of this theorem is:

∫_a^bf(x) dx = F(b) – F(a)

where ‘a’ and ‘b’ are the lower and upper limit of the integral, f(x) is the function about which the integral is to be calculated. 

The second fundamental theorem of calculus uses a well-defined function that is continuous over an open interval. 

d/dx ∫_a^x f(t) dt = f(x)

In the solution of this part of the theorem, ‘a’ can be known as a constant, as it does not affect the outcome of this function. This theorem can be used to denote definite integrals without the need for any constants. 

Newton-Leibniz Theorem

The Newton-Leibnitz theorem is the theorem that as its result gives us the formula using which we can calculate the differentiation of a definite integral of which limits are functions of a differential variable. This method in itself signifies the differentiation under an integral sign. 

A general definite integral is solved in the following way:

∫_a^bf(x) dx = F(b) – F(a)

where f(x) happens to be the derivative of the function F(x).

When the limits of the integral are completely different functions compared to the function of the integral sign, then the Newton-Leibniz method is applicable. 

d/dx [ ( ∫h(x)g(x) f(t) dt ] = f[g(x)] × g'(x) – f[h(x)] × h'(x)

In this formula, g(x) is the upper limit of the integral,

h(x) is the lower limit of integral

f(t) is the function

The solution of this theorem can also be given as:

d/dx [ ( ∫h(x)g(x) f(t) dt ] = function of upper limit × derivative of upper limit – the function of lower limit × derivative of the lower limit

This method of solving integrals is best used for the solution of integral transforms. 

Newton-Leibniz Formula Application

This theorem refers to the method using which one can find the derivative of an antiderivative. This theorem gives the formula using which a person can find the ‘nth order derivative of the product of two different functions. That also happens to be the reason why it is referred to as successive differentiation. 

For example;

Evaluate ∫0x cos t dt.

‘x’ in this problem is an independent entity. And, Cost is not a function of ‘x’.

Using the formula for the Newton-Leibniz theorem;

d/dx [ ( ∫h(x)g(x) f(t) dt ] = f[g(x)] × g'(x) – f[h(x)] × h'(x)

It can be observed that;

h(x) = x

g(x) = 0

Cost = f(t)

Therefore, the answer will be given as:

d/dx ∫0x cos t dt = cos x × (d/dx)(x) – cos 0 (d/dx) 0

Simplifying this entire equation, we get;

Cosx – 0

Cosx

This theorem can be used in the concepts of both integrations as well as differentiation. On the part of differentiation, it can be used to find the differential of a function to first order, second-order, and even to the nth order. Also, the concept of integration regarding this theorem is already known. 

Conclusion

There are different sets of formulas that are used to solve the integrals for the functions in questions. And that happens to be the reason why so many properties of the integrals are introduced. However, the normal properties of definite integrals do not apply to the problems in which the limit of integration is a function of a differential variable. Hence, this theorem is needed to solve the problems.