Integral involving trig substitution

In summary, trigonometric substitution is a technique used to solve integrals involving expressions with square roots of quadratic equations or the sum or difference of two squares. To use it, you need to identify the type of substitution needed and substitute the appropriate trigonometric expression. It is typically used for integrals involving these types of expressions or trigonometric functions. The most common substitutions used are <i>x = a sin&theta;</i>, <i>x = a cos&theta;</i>, <i>x = a tan&theta;</i>, and <i>x = a sec&theta;</i>. Some tips for using trigonometric substitution include checking if other techniques can be used, knowing common trigonometric identities, and choosing
  • #1
fran1942
80
0
Hello, I am trying to integrate 1/(x^2-1).

Apparently this can be solved by using trig substitution involving tan ?
Can someone please help me to understand how to go about it.

Thanks kindly for any help.
 
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  • #2
Let [itex]x= \sec \theta [/itex]
 
  • #3
Alternatively you can use partial fractions decomposition. You can also user ##x = \sin{(\theta)}## and multiply by (-1)/(-1).
 

1. What is trigonometric substitution in integrals?

Trigonometric substitution is a technique used to solve integrals that involve expressions containing square roots of quadratic equations or expressions containing the sum or difference of two squares.

2. How do you use trigonometric substitution in integrals?

To use trigonometric substitution, you first identify which type of trigonometric substitution is needed based on the form of the integral. Then, you substitute the appropriate trigonometric expression for the variable in the integral and use trigonometric identities to simplify the integral and solve for the variable.

3. When should trigonometric substitution be used?

Trigonometric substitution is typically used when the integral involves an expression with a square root of a quadratic equation or the sum or difference of two squares. It is also useful when the integral involves expressions with trigonometric functions.

4. What are the common trigonometric substitutions used in integrals?

The most common trigonometric substitutions used in integrals are:
- For expressions involving a square root of a quadratic equation: using the substitution x = a sinθ or x = a cosθ
- For expressions involving the sum or difference of two squares: using the substitution x = a tanθ or x = a secθ
- For expressions involving trigonometric functions: using the appropriate substitution based on the form of the integral.

5. Are there any tips for using trigonometric substitution in integrals?

Yes, here are some tips for using trigonometric substitution in integrals:
- Always check if the integral can be solved using other techniques such as u-substitution or integration by parts before using trigonometric substitution.
- Be familiar with common trigonometric identities and how to apply them to simplify the integral.
- Choose the appropriate substitution based on the form of the integral to make the integration process easier.

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