Introduction to the Differentiation under the Integral Sign

In calculus, the differentiation under the integral sign is a method of solving integrals that depend on a parameter other than that of the variable of integration. This method is also known as the Leibniz rule of integration. Let’s understand what is differentiation under the integral sign and how it is used to solve complex integrals easily. 

Understanding of the Leibniz Rule

The process of finding derivatives under an integral sign refers to the Leibniz rule which was first discovered by a German mathematician and scientist, Gottfried Leibniz. Since the method of integration involves the sum of all small areas of a curve between two points. The differentiation under the integral sign is the rate of change of an integral depending upon a parameter other than the variable of integration. Differentiation under the integral sign is also known as the Leibniz Rule. It is an operation or a rule to solve certain integrals. 

General Form of the Leibniz Rule

For a function $f(x,t)$ and its partial derivative $f_x(x,t)$ be continuous on x and t in some region of xt-plane such that $a(x)\le b(x), x_0 \le x\le x_1$. Also suppose that a(x) and b(x) functions are both continuous and have continuous derivatives for $x_0 \le x \le x_1$. Then, 

$\frac{d}{dx}\left(\int^{b(x)}_{a(x)} f(x,t) dt\right)=f(x,b(x)).\frac{d}{dx}b(x)-f(x,a(x)).\frac{d}{dx}a(x)+\int^{b(x)}_{a(x)} \frac{\partial}{\partial x}f(x,t)dt$

The above equation is the general form of the Leibniz integral rule. If the functions a(x) and b(x) are constants, then the above rule will become,

$\frac{d}{dx}\left(\int^b_a f(x,t) dt\right)=\int^b_a \frac{\partial}{\partial x}f(x,t)dt$

It is a special case of Leibniz rule which can be used to calculate differentiation under the integral sign involving constants as the interval points. Let’s understand how to calculate the integrals by using the Leibniz Rule. 

How do you calculate the differentiation under the integral sign?

The derivative under the integral sign can be calculated by using the Leibniz Integral Rule. This method seems tricky because it involves integral and differentiation both at the same time. Here is a step-by-step method of calculating differentiation using integrals. Follow these steps given below.

1. Identify the parameter and the coefficient of the integration. 
2. Differentiate both sides of the integral with respect to the parameter involved. 
3. Now calculate the integration with respect to the coefficient of the integration involved. 
4. Now substitute the values of the upper and lower bounds of the integral.
5. Simplify if needed.

Let’s solve the following example by using the Leibniz Rule of integral.

Leibniz Integral Rule example

Consider that we need to solve the integral, 

$\int^1_0 (2x+t^3)^2 dx$

In the first step, we will calculate the derivative of the integral with respect to t. So,

$\frac{d}{dt} \int^1_0 (2x+t^3)^2 dx=\int^1_0 \frac{\partial}{\partial t}(2x+t^3)^2 dx$

$\frac{d}{dt} \int^1_0 (2x+t^3)^2 dx=\int^1_0 2(2x+t^3)(3t^2)dx$

Simplifying, 

$\frac{d}{dt} \int^1_0 (2x+t^3)^2 dx=\int^1_0 (12t^2 x+6^t5)

Substituting the values of limits.

$\frac{d}{dt} \int^1_0 (2x+t^3)^2 dx=6t^2x^2+6t^5x|^{x=1}_{x=0}$

We get, 

$\frac{d}{dt} \int^1_0 (2x+t^3)^2 dx=6t^2+6t^5$

If the interval points are not given, you can use our indefinite integral calculator to solve such integrals.

Applications of the Leibniz Integral Rule

The differentiation under the integral sign or the Leibniz rule is an important and useful technique for solving complex integrals. It has many applications in engineering and science. Some of these applications are listed below.

1. It is used in different areas of physics to solve complex problems such as to find time derivatives of quantities like position, velocity and acceleration. 
2. In electrical engineering, the Leibniz rule analyzes signals and calculates electrical properties.
3. It plays an important role in statistics and probability to solve integrals related to the probability density functions, cumulative distribution functions and other statistical quantities.
4. It has many useful applications in quantum mechanics. It involves the study of the behavior of matter at atomic and subatomic levels. These calculations used the concept of differentiation and integration at the same time.

Conclusion

The differentiation under the integral sign is a technique of solving integrals depending on a parameter other than the coefficient of integration. It is calculated by differentiating the integrand with respect to the parameter involved and then solving the result by usual integration techniques such as by using a definite integral calculator. It has many applications in various fields of science like engineering, mechanics, electrical engineering, quantum mechanics etc.