What is the Substitution Method for Integrating a Rational Function?

Thread starter Physics197
Start date May 31, 2009
Tags

Integration Substitution

In summary, substitution integration is a method used in calculus to find integrals by substituting a variable with a new variable. It is used to solve integrals that cannot be solved using basic techniques and involves identifying the variable, choosing a new variable, rewriting the integral, solving for the new variable, and simplifying the expression. Common mistakes include not substituting the new variable back into the original integral and choosing the wrong variable. To improve skills, one can practice solving integrals using this method and review basic integration rules. Seeking help from a tutor or attending workshops can also be beneficial.

May 31, 2009

Physics197

Homework Statement

∫1/(x^2+2x+2) dx

Homework Equations

The Attempt at a Solution

u = x^2+2x+2
du = 2dx(x+1)

But I am left with an x and can not find the antiderviative

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May 31, 2009

Pengwuino

Gold Member

Try the substitution u = x+1.

Jun 1, 2009

Mark44

Mentor

Insights Author

Using the substitution suggested by Pengwuino, you should get
[tex]\int \frac{du}{u^2 + 1}[/tex]

Hopefully you know an antiderivative for this integral.

FAQ: What is the Substitution Method for Integrating a Rational Function?

What is substitution integration?

Substitution integration is a method used in calculus to find the integral of a function by substituting a variable with a new variable to simplify the expression.

How is substitution integration used?

Substitution integration is used to solve integrals that cannot be solved using basic integration techniques, such as u-substitution or integration by parts. It helps to simplify the expression and make it easier to integrate.

What are the steps to perform substitution integration?

The steps to perform substitution integration are as follows:

Identify the variable to be substituted.
Choose a new variable to replace the identified variable.
Rewrite the integral in terms of the new variable.
Solve for the new variable in terms of the original variable.
Substitute the new variable back into the original integral.
Simplify and solve the integral using basic integration rules.

What are some common mistakes made in substitution integration?

Some common mistakes made in substitution integration include forgetting to substitute the new variable back into the original integral, not simplifying the expression before solving the integral, and incorrectly choosing the new variable to substitute. It is important to double-check the steps and ensure that the final answer is in terms of the original variable.

How can one practice and improve their substitution integration skills?

The best way to practice and improve substitution integration skills is to solve a variety of integrals using this method. It is also helpful to review the basic integration rules and techniques to ensure a better understanding of the process. Additionally, seeking help from a tutor or attending workshops can also aid in improving substitution integration skills.