Techniques for Integrating Radical Functions?

In summary, the most common technique for integrating functions with radicals is trigonometric substitution. This involves using trigonometric identities to rewrite the integral in terms of trigonometric functions. Hyperbolic substitutions can also be used in some cases. These techniques are commonly taught in calculus courses.
  • #1
nickm4
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What are some techniques for integrating functions that are or contain radicals? I am familiar with trigonometric substitution. Are there any other "common" techniques or special functions that can be used to integrate these functions? or does trigonometric substitution basically take care of everthing? This is a general question, I don't have a specific function in mind.

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  • #2
Most common is to use trigonometric (or hyperbolic) substitutions.

[itex]sin^2(t)+ cos^2(t)= 1[/itex] so [itex]\sqrt{1- sin^2(t)}= cos(t)[/itex]

For example, to integrate [itex]\int \sqrt{1- x^2}dx[/itex], let x= sin(t). Then dx= cos(t) dt and [itex]\sqrt{1- x^2}= \sqrt{1- sin^2(t)}= cos(t)[/itex] so the integral becomes [itex]\int cos^2(t) dt[/itex] which can be integrated using the identity [itex]cos^2(t)= (1/2)(1+ cos(2t))[/itex].

For something like [itex]\sqrt{1+ x^2}[/itex] you can divide [itex]sin^2(t)+ cos^2(t)= 1[/itex] by cos(t) to get [itex]tan^2(t)+ 1= sec^2(t)][/itex]

To integrate [itex]\int \sqrt{9+ x^2}dx[/itex], let x= 3tan(t). Then [itex]dx= 3 sec^2(t) dt[/itex] and [itex]\sqrt{9+ x^2}= \sqrt{9+ 9tan^2(t)}= 3\sqrt{1+ tan^2(t)}= 3sec(t)[/itex] so the integral becomes [itex]\int (3 sec(t))(3sec^2(t)dt)= 9\int sec^3(t)dt[/itex]. That can be integrated by writing it as [itex]9\int dt/cos^3(t)= 9\int cos(t)dt/cos^4(t)= 9\int cos(t)dt/(1- sin^2(t))^2[/itex] and using the substitution u= sin(t).

But it is also true that [itex]cosh^2(t)- sinh^2(t)= 1[/itex] or [itex]sinh^2(t)+ 1= cosh^2(t)[/itex] so you could also use the substitution x= 3 sinh(x). Then dx= 3 cosh(x)dx and [itex]\int \sqrt{9+ x^2}dx= 9\int cosh^2(x) dx[/itex]

Just about any calculus text will devote at least a section, if not a chapter, to trig substitutions though hyperbolic substitution are less commonly covered.
 

1. What are radical functions?

Radical functions are mathematical expressions that contain a root, or radical, symbol (√). These functions involve taking the root of a number or variable, such as square roots, cube roots, or higher roots.

2. Why is it important to learn techniques for integrating radical functions?

Integrating radical functions is important because it allows us to solve complex mathematical problems involving roots. These techniques are essential in many fields of science and engineering, such as physics and electrical engineering.

3. What are some common techniques for integrating radical functions?

Some common techniques for integrating radical functions include substitution, completing the square, and using trigonometric identities. These techniques help simplify the expression and make it easier to perform integration.

4. How do I know which technique to use when integrating radical functions?

The choice of technique depends on the specific function being integrated. It is important to understand the properties of each technique and practice using them in different scenarios to determine the most suitable method for a given problem.

5. Are there any tips for successfully integrating radical functions?

One tip for successfully integrating radical functions is to be familiar with the rules of integration, such as the power rule and the substitution rule. It is also helpful to simplify the expression as much as possible before attempting to integrate, and to double-check the solution using differentiation to ensure accuracy.

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