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Using the UV formula to obtain the product of the two functions u and v is a straightforward way to discover the Integration. This formula for integrating UV may be used for algebraic equations, trigonometric ratios, and logarithmic quantities.
To express the provided integral, we use the differential of a product of functions to extend it. Hence, the term “integration by parts” or “the product rule of integration” refers to the process of integrating a UV formula. Let’s look at how the UV formula is integrated and what it may be used for.
The formula for Integration of UV
Integration by parts is a specialised technique of Integration that is beneficial in various situations, including when two functions are multiplied together. However, this type of Integration is also valuable on its own.
f the two functions are of the type u,v then the formula for the Integration of u and v may be written as follows:
∫ uv dx = u ∫ v dx – ∫ (u’ ∫ v dx) dx
Using the product rule of differentiation, we will construct the formula for the Integration of UV. We have two functions, u and v, and that y is the solution to the equation uv. When we use the product rule of differentiation, we will obtain the following results:
d/dx (uv) = u (dv/dx) + v (du/dx)
After some reorganization of the phrases, we have,
u (dv/dx) = d/dx (uv) – v (du/dx)
Perform the integration on both sides with regard to x,
∫ u (dv/dx) (dx) = ∫ d/dx (uv) dx – ∫ v (du/dx) dx
⇒ ∫u dv = uv – ∫v du
As a result, the Integration of the UV formula may be obtained.
Solved example
Q: Find the integral of x3.logx
Solution:
Here u = logx and dv = x3 dx
du = 1/x dx and v = x4 /4 + C
Using the integration of uv formula ∫u.dv = uv- ∫v du we get
∫x3 log x dx= log x. (x 4/4) – ∫(x4/4)(1/x)dx
= log x. (x 4/4) -(1/4) ∫( (x 4)(1/x)dx
= log x. ( (x 4/4)) -(1/4) ∫x3dx
= ( (x 4/4))log x – (1/4) ( (x 4/4))+C
= (x 4/4)) log x- (x4/16)+ C
