Integration of UV Formula

Integration of uv formula is a convenient means of finding the integration of the product of the two functions u and v. There are two forms of this formula: ∫ uv dx = u ∫ v dx - ∫ (u' ∫ v dx) dx (or) ∫ u dv = uv - ∫ v du.

Further, the two functions used in this integration of uv formula can be algebraic expressions, trigonometric or logarithmic functions. We expand the differential of a product of functions and express the given integral in terms of a known integral. Thus uv rule of integration is also known as integration by parts or the product rule of integration. Let's learn the integration of uv formula and its applications.

What is Integration of UV Formula?

The integration of uv formula is a special rule of integration by parts. Here we integrate the product of two functions. If u(x) and v(x) are the two functions and are of the form ∫u dv, then the Integration of uv formula is given as:

• ∫ uv dx = u ∫ v dx - ∫ (u' ∫ v dx) dx
• ∫ u dv = uv - ∫ v du

In the first formula, u is the first function and v is the second function. On the other hand, in the second formula, u is the first function and dv is the second function.

To decide which of the two given functions is u, we have to use LIATE (or) ILATE rule whose abbreviation is given below. Here, whichever of the given two functions first appears in this rule's list is the first function.

• L - Logarithmic function
• I - Inverse trigonometric function
• A - Algebraic Function
• T - Trigonometric function
• E - Exponential Function

Integration of UV Formula

We follow the following simple quick steps to find the integral of the product of two functions:

• Identify the functions 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 (= u')
• Integrate v: ∫v dx
• Key in the values in the formula ∫u · v dx = u ∫v dx- ∫(u' ∫(v dx)) dx
• Simplify and solve.

Derivation of Integration of UV Formula

We will derive 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 sides with respect to x,

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

⇒ ∫u dv = uv - ∫v du

Hence, the product rule of integration is derived.

Let us try out a few examples to understand better how to apply the uv rule of integration.

Solved Examples Using Product Rule of Integration

Example 1: Find the integral of x · sin x.

Solution:

Here u = x and dv = sin x dx

du = dx and v = ∫ sin x dx = - cos x dx

Using the product rule of integration, ∫u dv = uv- ∫v du we get

∫x sin x dx = x (- cos x) - ∫(- cos x dx)

= -x cos x + sin x + C

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

Example 2: Find the integral of x² · ln x.

Solution:

Here u = ln x and dv = x² dx

du = 1/x dx and v = ∫x² dx = x³ /3

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

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

= ln x (x ³/3) -(1/3) ∫(x³)(1/x)dx

= ln x (x³/3) -(1/3) ∫x² dx

= (x³/3)ln x - (1/3) (x³ /3)+C

= (x³/3) ln x - (x³/9)+ C

Answer: ∫x² ln x = (x³/3) ln x - (x³/9)+ C

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

Solution:

Here u = x and dv = e^x dx.

du = dx and v = ∫ e^x dx = e^x.

Using the uv rule of integration: ∫u dv = uv- ∫v du, we get

∫xe^x dx = x e^x - ∫ e^x dx

= xe^x - e^x + C

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