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Updated on 31st July, 2023 , 5 min read
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
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,
The following steps helps to find the integral of the product of two functions:
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.
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.
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.
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.
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
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
Solution:
Here u = xand dv = exdx.
du = dx and v = ex
Using the integration of uv formula ∫u.dv = uv- ∫v du, we get
∫xexdx = x ex - ∫ ex dx
= xex- ex+ C
Answer: Thus, integral of xexdx=xex- ex+C
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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By - Nikita Parmar 2024-09-06 10:59:22 , 6 min readThe UV rule of integration, also known as integration by parts, is a method used to integrate the product of two functions.
The UV rule is used when the integrand is a product of two functions that cannot be integrated by simpler techniques such as substitution.
The formula for the UV rule is ∫ u dv = u v - ∫ v du, where u and v are functions of x.
Choose u in a way that the derivative du/dx is simpler than u itself, and choose dv in a way that the integral ∫ dv is easier than v itself.
Yes, the UV rule can be applied multiple times if necessary to fully evaluate the integral.
No, the UV rule is not applicable to all functions. It is typically used for products of functions that are difficult to integrate using other techniques.
Common examples of u functions include logarithmic functions, inverse trigonometric functions, and algebraic functions.
Common examples of dv functions include exponential functions, trigonometric functions, and algebraic functions.
When evaluating a definite integral using the UV rule, you must apply the rule to find the antiderivative first, and then evaluate the expression over the given bounds. When evaluating an indefinite integral, you must include the constant of integration.
The UV rule is used in various fields such as physics, engineering, and economics to solve problems that involve integrals of products of functions, such as finding the work done by a variable force or the growth rate of a population.