Integration By Parts Formula

About Integration By Parts Formula

Integrals are usually calculated for functions that have differentiation formulas. Integration by parts, also known as partial integration, is an extra technique for determining the integration of the product of functions. It converts the product of functions into integrals for which a solution may be quickly computed.
Because some inverse trigonometric and logarithmic functions lack integral formulae, we can utilise the integration by parts formula instead. We'll look at the proof, the graphical representation, applications, and integration by parts examples here.

What do you mean by Integration by Parts:

Integration by parts is a technique for combining the output of two or more functions. The two functions to be integrated are of the form f(x) and g(x) is ∫f(x).g(x). As a result, it's known as a product rule of integration. The first function, f(x), is chosen because its derivative formula exists, while the second function, g(x), is chosen because an integral of such a function exists.

  • ∫ f(x).g(x).dx = f(x) ∫ g(x).dx − ∫ (f′(x) ∫g(x).dx).dx+C

(First Function x Second Function) integration = (First Function) x (Integration of Second Function) - Integration of Second Function (Differentiation of First Function x Integration of Second Function).

The formula is divided into two parts in the integration by parts method, and we can see the derivative of the first function f(x) in the second part and the integral of the second function g(x) in both parts. These functions are frequently expressed as 'u' and 'v' for clarity. The integration of the uv formula using the 'u' and 'v' notation is:

  • ∫ u dv = uv - ∫ v du.

Integration By Parts Formula

The integral of the product of two distinct types of functions, such as logarithmic, inverse trigonometric, algebraic, trigonometric, and exponential functions, is found using the integration by parts formula. The integral of a product is calculated using the integration by parts formula. uv, u(x), and v(x) can be chosen in any sequence in the product rule of differentiation when we differentiate a product. However, when utilizing the integration by parts formula, we must first determine which of the following functions occurs first in the following order before assuming it is u.

  1. Logarithmic (L)
  2. Inverse trigonometric (I)
  3. Algebraic (A)
  4. Trigonometric (T)
  5. Exponential (E)

This can be remembered using the rule LIATE. Note that this order can be ILATE as well. For example, if we have to find ∫ x ln x dx (where x is an algebraic function and ln is a logarithmic function), we will choose ln x to be u(x) as in LIATE, the logarithmic function appears before the algebraic function. It is defined in two ways:

  • ∫ f(x).g(x).dx = f(x) ∫ g(x).dx− ∫(f′(x) ∫g(x).dx).dx+C
  • ∫ u dv = uv - ∫ v du.

Get a List of all Maths Formulas on one page

Download the pdf of Integration By Parts Formula

Integration By Parts Formula
Integration By Parts Formula

Related Links

Frequently Asked Questions on Integration By Parts Formula