Is there a chain rule for integration?
Introduction
In derivatives, we have a chain rule to evaluate derivatives of complex functions. Same as derivatives, there is a chain rule in integration known as the reverse chain rule. This article will help you to understand the reverse chain rule and the difference between the chain rule in derivative and integration.
Reverse Chain Rule
The reverse chain rule is a technique of finding integration of a function whose derivative is multiplied with it. Since the chain rule is used for derivatives to calculate derivative of complex functions or the function in combination form. It is a technique that allows us to find derivatives.
We can do the reverse of chain rule to integrate complicated functions where the function and its derivative appear in a combined form. The reverse chain rule combines these two parts of the function and integrates it directly. This rule can also be called the “substitution Rule", or the “U-Substitution Rule".
Reverse Chain Rule Formula
Suppose for a function y = fgx the chain rule formula can be written as,
$$\frac{dy}{dx}\;=\;\frac{dy}{du}\;\times\;\frac{du}{dx}$$
Where u = g(x). Now the reverse chain rule can written as,
$$∫ f(g(x))g'(x)dx\;=\;∫f(u)du$$
Where u = g(x) and u' = g'(x).
U-substitution helps us to find the integral of functions y = f(g(x))g’(x) in which outside function u’ = g’(x) is the derivative of the inside function u = g(x). This chain rule for integration is known as the U-substitution rule because we can substitute the derivative of a function with ‘u’. You can also use chain rule calculator.
How to apply Substitution Rule?
Consider a function y = f (g(x)) g’(x) that can be integrated by using following steps:
- Determine the function and its derivative and substitute it as u = g(x).
- Find the derivative of u such that du = g'(x)dx.
- Rewrite the integral in terms of u and du.
- Now integrate the function with respect to u.
- After integrating the function, replace u by g(x).
Let’s discuss an example to apply the u-substitution rule.
Example:
Suppose an indefinite integral ∫ x sin x2dx. Use the u-substitution method to find antiderivatives.
solution:
Given integral is,
$$∫\;x\;sin\;x^2dx$$
Suppose,
$$x^2\;=\;u $$
And
$$2x\;dx\;=\;du$$
From the above integral, multiply and divide by 2 to form a derivative of u.
$$∫\;x\;sin\;x^2dx\;=\;\frac{1}{2}∫2x\;sin\;x^2\;dx $$
Now, writing the integral in the form of u,
$$∫\;x\;sin\;x^2dx\;=\; \frac{1}{2}\;∫\;du $$
Integrating with respect to u,
$$∫\;x\;sin\;x^2dx\;=\; -\frac{1}{2} cos\;u\;+\;c$$
Replacing x2 = u ,
$$∫\;x\;sin\;x^2dx\;=\; -\frac{1}{2} cos\;x^2\;+\;c$$
Example:
Solve the following integral by reverse chain rule.
$$∫(2x\;+\;1)^3dx$$
Solution:
Let
$$2x\;+\;1\;=\;u$$
And,
$$2dx\;=\;du$$
From the above integral, multiply and divide by 2 to form a derivative of u.
$$∫(2x\;+\;1)^3dx\;=\;\frac{1}{2}∫u^3du$$
Now integrating with respect to u.
$$∫(2x\;+\;1)^3\;dx\;=\;\frac{1}{2}\;\times\;\frac{u^4}{4}\;+\;c$$
$$∫(2x\;+\;1)^3dx\;=\;\frac{u^4}{8}\;+\;c$$
Substituting the value of u
$$ ∫\;(2x\;+\;1)^3\;dx\;=\;\frac{(2x\;+\;1)^4}{8}\;+\;c $$
When can you Reverse the Chain Rule?
The reverse chain rule is a method of finding an antiderivative of a function. It helps to evaluate integration with a direct substitution method. It made it easier to find integration of complex functions.
You can reverse the chain rule when you have to find an antiderivative of a function that is in combined form with its derivative. For example if a function is,
$$y\;=\;f(g(x))g’(x)$$
The integration becomes difficult when a function appears with its derivative. So, we can do the reverse chain rule to integrate this function that appears with its derivative.
Difference between Chain Rule and Reverse Chain Rule
The derivative and integration both are fundamental concepts of calculus. There are many techniques to evaluate them. The chain rule and reverse chain rule are one of those techniques that allow us to solve derivatives and antiderivatives directly. Here we will discuss the difference between these techniques.
| Chain Rule |
Reverse Chain Rule |
| It is a technique of finding derivatives of a function that appears in a combination of two functions. |
It is a technique of finding an antiderivative of a function that appears with its derivative. |
The chain rule formula for a function y = f(g(x)) is,
$$\frac{dy}{dx}\;=\;\frac{dy}{du}\;\times\;\frac{du}{dx}$$
Where u = g(x) |
The reverse chain rule formula for a function
y = f(g(x)) g’(x) is,
$$∫f(g(x))g'(x)dx\;=\;∫f(u)du$$
Where u = g(x) |
| It calculates the rate of change of a function with respect to another function. |
It calculates the antiderivative of a function by using its derivative. |
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Related:Is Chain Rule same as Product Rule? - Difference & Comparison
FAQ’s
Does Chain Rule exist in Integration?
Yes. There is a chain rule in integration also that is the inverse of chain rule in derivatives. It is used to solve those integrals in which the function appears with its derivative. This rule is also known as the substitution method.
How many Rules are there in Integration?
The fundamental theorem of calculus defines two rules to solve integration. One rule is to find the derivative of indefinite integrals and the second is to solve definite integrals. These are,
- d / dx x∫a f(t)dt = f(x) (derivative of indefinite integrals)
- b∫a f(t) dt = F(b) - F(a) (integration of definite integrals)
Is there a Chain Rule in Integration?
Yes, there is a technique of finding integration by using chain rule in integration. It is known as reverse chain rule or u-substitution or substitution rule. It helps to find integration of a complex function with a direct and easy method.