What is UV Rule of Integration with Solved Examples

UV Rule of Integration Overview

The UV rule of Integration formula provides a convenient method to compute the integral of the product of two functions, u and v. This formula can be applied to various functions, including algebraic expressions, trigonometric ratios, and logarithmic functions. By expanding the differential of the product of functions, we can express the given integral in terms of a known integral. Hence, the Integration of UV formula is also known as Integration by Parts or the Product Rule of Integration. Understanding the integration of the UV formula and its applications can be highly beneficial in solving complex integration problems.

Also Read: ILATE

What is the UV Rule of Integration?

The UV Rule of Integration is a specific technique used in integration to evaluate the integral of the product of two functions. Suppose u(x) and v(x) are two functions in the form of ∫u dv, then the UV Rule of Integration formula can be applied as follows:

∫ uv dx = u ∫ v dx - ∫ (u' ∫ v dx) dx

Alternatively, the UV Rule of Integration formula can also be expressed as:

∫ u dv = uv - ∫ v du

Here, 

• u = function of u(x)
• dv = variable dv
• v = function of v(x)
• du = variable du

UV Rule of Integration

The following steps helps to find the integral of the product of two functions:

• Identify the function u(x) and v(x). Choose u(x) using the LIATE rule: whichever first comes in this order: Logarithmic, Inverse, Algebraic, Trigonometric, or Exponential function.
• Find the derivative of u: du/dx
• Integrate v: ∫v dx
• Key in the values in the formula ∫u.v dx = u. ∫v.dx- ∫( ∫v.dx.u'). dx
• Simplify and solve.

UV Rule of Integration: Derivation

Deriving the integration of uv formula using the product rule of differentiation. Let us consider two functions u and v, such that y = uv. On applying the product rule of differentiation, we will get,

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

Rearranging the terms, we have,

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

Integrate on both the sides with respect to x,

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

⇒∫u dv = uv - ∫v du

Hence, the integration of the UV formula is derived.

How Does the UV Rule of Integration Work?

To understand the UV rule of integration, let's consider the product of two functions, u(x) and v(x), and attempt to integrate them. We begin by applying the product rule of differentiation to the product of u(x) and v(x):

(uv)' = u'v + uv'

Now we can rearrange this equation to get:

uv' = (uv) - u'v

Integrating both sides of the equation with respect to x, we get:

∫uv'dx = ∫(uv - u'v)dx

Using the distributive property of integration, we can break this down into two integrals:

∫uv'dx = ∫uvdx - ∫u'vdx

And this is the UV rule of integration.

How to Use the UV Rule of Integration

Using the UV rule of integration involves a few steps:

Step 1:Identify the functions u(x) and v(x) in the integral to be evaluated.

Step 2: Take the derivative of u(x) and the integral of v(x).

Step 3: Substitute the values of u(x), u'(x), v(x), and ∫v(x)dx into the UV rule of integration formula.

Step 4:Simplify the equation and evaluate the integral.

Let's look at an example to see how the UV rule of integration works in practice.

Example: Evaluate the integral of xcos(x)dx

Step 1:Identify the functions u(x) and v(x).

Let u(x) = x and v'(x) = cos(x)

Step 2:Take the derivative of u(x) and the integral of v(x).

u'(x) = 1 and ∫v(x)dx = sin(x)

Step 3:Substitute the values into the UV rule of integration formula.

∫xcos(x)dx = x sin(x) - ∫sin(x)dx

Step 4: Simplify the equation and evaluate the integral.

∫xcos(x)dx = x sin(x) + cos(x) + C

Using the UV rule of integration, we were able to evaluate the integral of xcos(x)dx.

UV Rule of Integration: Solved Example

Example 1: Find the integral of x.Sinx.

Solution:

Here u = x and dv = sin x dx

du = dx and v = ∫sinx dx= - cos x dx

Using the uv formula ∫u.dv = uv- ∫v du we get

∫x sinx dx = x. (-cos x) - ∫(-cos x dx)

= -x cos x - (-sin x) + C

= -x cos x + sin x + C

Answer: ∫x.sinx.dx = sin x - x cos x + C

Example 2: Find the integral of x².logx

Solution:

Here u = logx and dv = x²dx

du = 1/x dx and v = x³ /3 + C

Using the integration of uv formula ∫u.dv = uv- ∫v du we get

∫x² log x dx= log x. (x ³/3) - ∫(x³/3)(1/x)dx

= log x. (x ³/3) -(1/3) ∫(x³)(1/x)dx

= log x. (x ³/3) -(1/3) ∫x²dx

= (x³/3)log x - (1/3) (x³ /3)+C

=(x³/3) log x- (x³/9)+ C

Answer: ∫x²logx = (x³/3) log x- (x³/9)+ C

Example 3: Find the integral of xe^x dx.

Solution:

Here u = xand dv = e^xdx.

du = dx and v = e^x

Using the integration of uv formula ∫u.dv = uv- ∫v du, we get

∫xe^xdx = x e^x - ∫ e^x dx

= xe^x- e^x+ C

Answer: Thus, integral of xe^xdx=xe^x- e^x+C

Example 4:  Find the integral of x³.logx

Solution:

Here u = logx and dv = x³dx

du = 1/x dx and v = x⁴/4 + C

Using the integration of uv formula ∫u.dv = uv- ∫v du we get

∫x³log x dx= log x. (x⁴/4) – ∫(x⁴/4)(1/x)dx

= log x. (x⁴/4) -(1/4) ∫( (x⁴)(1/x)dx

= log x. ( (x⁴/4)) -(1/4) ∫x3dx

= ( (x⁴/4))log x – (1/4) ( (x⁴/4))+C

Answer:  (x⁴/4)) log x- (x⁴/16)+ C

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UV Rule of Integration: Things to Remember

1. The UV rule is also known as integration by parts.
2. The formula for the UV rule is: ∫ u dv = u v - ∫ v du.
3. The goal of the UV rule is to transform a difficult integral into an easier one.
4. When selecting u and dv, it is important to choose u in a way that the derivative du/dx is simpler than u itself, and dv in a way that the integral ∫ dv is easier than v itself.
5. When applying the UV rule, differentiate u and integrate dv to find du and v, respectively.
6. After applying the UV rule, you may need to apply it multiple times, or use other integration techniques to fully evaluate the integral.
7. Common examples of u functions include logarithmic functions, inverse trigonometric functions, and algebraic functions.
8. Common examples of dv functions include exponential functions, trigonometric functions, and algebraic functions.
9. Be careful when evaluating indefinite integrals to include the constant of integration, denoted by C.
10. The UV rule is often used in calculus courses to evaluate integrals, and is an important tool for solving real-world problems in fields such as physics, engineering, and economics.