Anti-derivative is a dedicated concept of mathematical equations. Calculating an anti-derivative is a difficult process that requires an understanding of various respects. However, in the case of multiplication between two functions, calculating the anti-derivatives can become a more comprehensive and difficult process that requires the use of the concept of integration by parts.

As evident from its name, integration by parts helps in the calculation of the result of the product of functions. The concept delivers a dedicated formula that can help in making the process of calculating the correct result of multiplication between two functions and their derivatives complete hassle-free and convenient. Two functions are multiplied together and calculation of its anti-derivative through the integration of parts formula becomes easy. The idea behind using this concept and formula is to solve a function without dividing it into two parts. Two or more comprehensive functions multiplied together can be calculated without breaking them altogether.

## 1. Calculation of integration of parts:

Formula for calculating multiplication of two functions through integration of parts is as follows:

∫f(x).g(x)dx = f(x)∫g(x)dx−∫f′(x).(∫g(x)dx)dx

Here f is the first function and g is the second function respectively. x is the variable.

f′(x) is the differential coefficient of the first function

## 2. Steps for the calculating product of two functions through the use of the integration by parts formula:

The steps for use of the above-given formula and calculation of the result of multiplication between two functions of integration of parts are listed below:

- Find the dedicated value of f(x) and g(x).
- Differentiate the value of f(x) that derives the result of f’(x)
- Now the next step involves the integration of g(x). This delivers the value of ∫g(x) dx
- After calculation of all the necessary values, these are required to be inputted into the above-given formula.
- After all the values have been calculated and inserted into the formula for integration of parts, The formula and calculation become simplified. The correct result can be easily calculated within minutes through the use of the formula as given through integration by parts.

For example – ∫ xsin(x) dx

Here f(x) = x and g(x) = sin(x)

Using this formula for integration of parts ∫f(x).g(x)dx = f(x)∫g(x)dx−∫f′(x).(∫g(x)dx)dx

∫g(x) = ∫sin(x) = -cos(x)

Inputting these values in the equation:

∫ xsin(x) dx = x(-cos(x)) – ∫ -cos(x) dx

= -xcos(x) + ∫cos(x) dx

= -xcos(x) + sin(x) + c

The antiderivative has been calculated through the use of integration of parts method and formula.

## 3. Ilate Rule:

According to the Ilate Rule, In the case of the product of two functions, the integral of the functions is taken where the left one is considered as the first function and the right term is considered to be the second function. In the above example, ∫ xsin(x) dx

x is the first function and sin(x) is the second function. The rule is to select the function in such a way so that the derivative of the function can be calculated with ease.

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