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. Further, the two functions used in this integration of uv formula can be algebraic expressions, trigonometric ratios, or logarithmic functions. We expand the differential of a product of functions and express the given integral in terms of a known integral. Thus integration of uv formula 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

where,

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

Integration of UV Formula

We follow the following simple quick steps 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.

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 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 uv formula is derived.

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

Solved Examples Using Integration of UV Formula

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 = x² dx

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 = x and dv = e^xdx. 

du = dx and v = e^x

Using the integration of uv formula  ∫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