Integration by Parts

Brook Taylor, who is also credited with proposing the famous Taylor’s Theorem, first proposed the concept of integration by parts in 1715. It is customary to calculate integrals for functions that can be represented by differentiation formulas. Here When attempting to integrate the product of functions, an additional technique known as integration by parts can be used. It is also referred to as partial integration in some circles. It converts the integration of the product of functions into integrals, for which a solution can be easily computed, thereby simplifying the integration process.

The integral formulas for some inverse trigonometric functions and logarithmic functions are missing, and in these cases, we can use the integration by parts formula to solve the problem.

Integration by parts is a technique for bringing together the results of two or more functions. In this case, the two functions to be integrated, f(x) and g(x), have the form ∫f(x).g(x). As a result, it can be referred to as a product rule of integration. A function f(x) is chosen from among the two functions in such a way that its derivative formula exists, and a function g(x) is chosen in such a way that an integral of such a function exists from among the two functions.

   ∫f(x).g(x) dx = f(x) ∫ g(x).dx – ∫ (f ‘)(x) ∫ g(x).dx

It is calculated as follows: integration of (first function x second function) = (first function) x (Integration of Second Function – Integration of (Differentiation of First Function x Integration of Second Function).

Integration by parts is a method in which the formula is divided into two parts, and we can observe the derivative of the first function f(x) in the second part and the integral of the second function g(x) in both parts when the formula is divided. For the sake of simplicity, these functions are commonly denoted by the letters ‘u’ and ‘v,’ respectively. Integration of uv formula using the notation of ‘u’ and “v” is represented by the following:

          ∫u dv = uv – ∫ v du

Integration by parts formula

It is necessary to use the integration by parts formula in order to find the integral of the product of two different types of functions, such as logarithmic, inverse trigonometric, logarithmic trigonometric, algebraic, trigonometric, and exponential functions. When attempting to calculate the integral of a product, the integration by parts formula is employed. In the product rule of differentiation, which is used to differentiate a product, the variables uv, u(x), and v(x) can be chosen in any combination. However, when using the integration by parts formula, in order to select the first function u(x), we must first determine which of the following functions appears first in the following order, and then we must assume that function to be u.

The logarithmic scale (L)

Inverse trigonometric functions (I)

Algebraic expressions (A)

Trigonometric functions (T)

Exponential growth (E)

The rule LIATE can be used to help you remember this. It should be noted that this order can also be ILATE. When solving a problem such as finding ∫ln x dx (where x is an algebraic function and ln is a logarithmic function), we will choose ln x to be u(x) because, in LIATE, the logarithmic function appears before the algebraic function. There are two ways in which the integration by parts formula can be defined. We can use either of them to combine the results of two functions into a single product.

Application of integration by parts

The application of this formula for integration by parts is limited to functions or expressions for which there are no integration formulas available at the time of application. In this section, we will attempt to incorporate the integration by parts formula and to derive the integral. Integral answers are not available for logarithmic functions or inverse trigonometric functions, among other things. Consider the following problem: Let us try to solve and find the integration of log x and tan-1x.

Integration of logarithmic function

∫ log x.dx = ∫ logx.1.dx

= log x. ∫1.dx – ∫ ((log x)’.∫ 1.dx).dx

=logx.x -∫ (1/x .x).dx

=x log x – ∫ 1.dx

=x log x – x + C

Integration of inverse trigonometric function 

∫ tan-1x.dx = ∫tan-1x.1.dx

= tan-1x.∫1.dx – ∫((tan-1x)’.∫ 1.dx).dx

= tan-1x. x – ∫(1/(1 + x2).x).dx

= x. tan-1x – ∫ 2x/(2(1 + x2)).dx

= x. tan-1x – ½.log(1 + x2) + C

Conclusion

When two functions are multiplied together, Integration by Parts is a special method of integration that is often useful. However, it can be useful in a variety of other situations.  Integration by parts is a technique for bringing together the results of two or more functions. In this case, the two functions to be integrated, f(x) and g(x), have the form ∫f(x).g(x). As a result, it can be referred to as a product rule of integration.

Integration by parts is a method in which the formula is divided into two parts, and we can observe the derivative of the first function f(x) in the second part and the integral of the second function g(x) in both parts when the formula is divided.

It is necessary to use the integration by parts formula in order to find the integral of the product of two different types of functions, such as logarithmic, inverse trigonometric, logarithmic trigonometric, algebraic, trigonometric, and exponential functions.

The application of this formula for integration by parts is limited to functions or expressions for which there are no integration formulas available at the time of application.