Integration By Parts - Formula, Proof, Derivation, Examples, FAQs

Integration by Parts

The idea of integration by parts in calculus was proposed in 1715 by Brook Taylor, who also proposed the famous Taylor's Theorem. Generally, integrals are calculated for functions for which differentiation formulas exist. Here integration by parts is an additional technique used to find the integration of the product of functions and it is also referred to as partial integration. It changes the integration of the product of functions into integrals for which a solution can be easily computed.

Some of the inverse trigonometric functions and logarithmic functions do not have integral formulas, and here we can make use of integration by parts formula which is also popularly known as uv integration formula. Here we shall check the derivation, the graphical representation, applications, and examples of integration by parts.

1.	What Is Integration by Parts?
2.	Integration by Parts Formula
3.	Integration by Parts Formula Derivation
4.	Visualizing Integration by Parts
5.	Applications of Integration by Parts
6.	Formulas Related to Integration by Parts
7.	FAQs on Integration by Parts

What is Integration by Parts?

Integration by parts is used to integrate the product of two or more functions. The two functions to be integrated f(x) and g(x) are of the form ∫f(x)·g(x). Thus, it can be called a product rule of integration. Among the two functions, the first function f(x) is selected such that its derivative formula exists, and the second function g(x) is chosen such that an integral of such a function exists.

∫ f(x)·g(x)·dx = f(x) ∫ g(x)·dx - ∫ [(f'(x) ∫ g(x)·dx)·dx] + C

The 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).

In the integration by parts, the formula is split 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 the parts. For simplicity, these functions are often represented as 'u' and 'dv' respectively. The uv integration formula using the notation of 'u' and 'dv' is:

∫ u dv = uv - ∫ v du.

Integration By Parts Formula

The integration by parts formula is used to find the integral of the product of two different types of functions such as logarithmic, inverse trigonometric, algebraic, trigonometric, and exponential functions. The integration by parts formula is used to find the integral of a product. In the product rule of differentiation where we differentiate a product uv, u(x), and v(x) can be chosen in any order. But while using the integration by parts formula, for choosing the first function u(x), we have to see which of the following function comes first in the following order and then assume it as u.

• Logarithmic (L)
• Inverse trigonometric (I)
• Algebraic (A)
• Trigonometric (T)
• Exponential (E)

This can be remembered using the rule LIATE. Note that this order can be ILATE formula 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. The integration by parts formula is defined in two ways. We can use either of them to integrate the product of two functions.

Integration By Parts Formula Derivation

The proof of integration by parts can be obtained from the formula of the derivative of the product of two functions. Thus, the integration by parts formula is also known as the product rule of integration.

Let us derive the integration by parts formula using the product rule of differentiation. Consider two functions u and v. Let their product be y. i.e., y = uv. Applying the product rule of differentiation, we get

d/dx (uv) = u (dv/dx) + v (du/dx)

This can be written as:

u (dv/dx) = d/dx (uv) - v (du/dx)

Integrating on both sides with respect to x,

∫ u (dv/dx) (dx) = ∫ d/dx (uv) dx - ∫ v (du/dx) dx

By cancelling the terms,

∫ u dv = uv - ∫ v du

Hence the integration by parts formula is derived.

Visualizing Integration by Parts

Consider a parametric curve (x, y) = (f(θ), g(θ)). Let us consider this curve to be integrable and a one-to-one function. The integration by parts represents the area of the blue region from the below curve. Let us first consider the areas of the blue region and the yellow regions distinctly.

Consider the curve along the y-axis we have the function x(y) and across the limits [y1, y2]. Also we can consider the curve along the x-axis and have the function y(x) across the limits [x1, x2].

Area of the yellow region = ∫y2y1 x(y)·dy

Area of the blue region = ∫x2x1 y(x)·dx

The total area of these two regions is equal to the area of the larger rectangle minus the area of the smaller rectangle.

∫y2y1 x(y)·dy + ∫x2x1 y(x)·dx = [x·y(x)]x2x1

Without the definite integrals, it can be written as.

∫ y·dx+ ∫ x·dy = xy

∫x·dy = xy - ∫ y·dx

Further, this can be modified to obtain the integration by parts formula.

∫f(x)·g(x)·dx = f(x)·∫ g(x)·dx - ∫(f'(x) · ∫ g(x)·dx) ·dx

Applications of Integration by Parts

The application of this formula for integration by parts is for functions or expressions for which the formulas of integration do not exist. Here we try to include this formula of integration by parts and try to derive the integral. For logarithmic functions and for inverse trigonometry functions there are no integral formulas. Let us try to solve and find the integration of log x and tan-1x.

Integration of Logarithmic Function

∫ log x·dx = ∫ log x.1·dx

= log x. ∫1·dx - ∫ ((log x)'.∫ 1·dx)·dx

= log x·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

Formulas Related to Integration by Parts

The following formulas have been derived from the integration by parts formula and are helpful in the process of integration of various algebraic expressions.

• ∫ ex(f(x) + f'(x))·dx = exf(x) + C
• ∫√(x2 + a2)·dx = ½ . x·√(x2 + a2)+ a2/2· log|x + √(x2 + a2)| C
• ∫√(x2 - a2)·dx =½ x·√(x2 - a2) - a2/2· log|x +√(x2 - a2) | C
• ∫√(a2 - x2)·dx = ½ x·√(a2 - x2) + a2/2· sin-1 x/a + C

☛ Related Articles:

• Calculus
• Differentiation and Integration Formulas
• Differential Equations
• Integration Formulas

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