How Does the Substitution Rule Work?

In summary, the Substitution Rule, also known as the Chain Rule, is a method used in calculus to solve integrals by rewriting them in terms of the inner function and its derivative. It is used to simplify integrals by replacing the variable with a new variable that is easier to integrate. The steps for using the Substitution Rule include identifying the inner function and its derivative, letting the inner function be the new variable, and rewriting the integral in terms of the new variable. The Substitution Rule is important in calculus because it allows for the integration of complex functions and makes it easier to find the area under a curve. It can be used for both indefinite and definite integrals, with the limits of integration also being adjusted for definite integrals
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Homework Statement


If u=g(x) is a differentiable function whose range is an interval I and f is continuous on I, then Integral f(g(x)g'(x)dx=Integral f(u)du.


Homework Equations





The Attempt at a Solution

 
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3. The Attempt at a Solution ?
 

What is the Substitution Rule?

The Substitution Rule, also known as the Chain Rule, is a method used in calculus to solve integrals. It states that if a function is composed of two or more functions, the integral of the function can be rewritten in terms of the inner function and its derivative.

How is the Substitution Rule used?

The Substitution Rule is used to simplify integrals by replacing the variable with a new variable that is easier to integrate. This new variable is obtained by solving for the inner function and its derivative.

What are the steps for using the Substitution Rule?

The steps for using the Substitution Rule are as follows:
1. Identify the inner function and its derivative
2. Let the inner function be the new variable
3. Replace the original variable and its derivative with the new variable
4. Rewrite the integral in terms of the new variable
5. Integrate the new function
6. Rewrite the solution in terms of the original variable

Why is the Substitution Rule important in calculus?

The Substitution Rule is important in calculus because it allows for the integration of complex functions that cannot be solved using basic integration techniques. It simplifies the integration process and makes it easier to find the area under a curve.

Can the Substitution Rule be used for definite integrals?

Yes, the Substitution Rule can be used for both indefinite and definite integrals. For definite integrals, the limits of integration must also be adjusted to match the new variable.

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