Evaluating the Integral Using Trig Substitution

In summary, the conversation is about evaluating an integral using trig substitution. The integral has been reduced to ∫ (tanz)^2 where z is found using a triangle. The person is stuck on integrating tangent squared and has tried using the trig identity (secx)^2 - 1. A hint is given to differentiate tan(z) and the person realizes their mistake and gets (sec(z))^2. They are reminded to find an anti-derivative for sec^2 and then realize they can use this to find an anti-derivative for tan^2(z)=sec^2(z)-1. The conversation ends with the person thanking for the help.
  • #1
cmantzioros
29
0
The question is to evaluate the integral in the attachment.

Using trig substition, I've reduced it to ∫ (tanz)^2 where z will be found using the triangle. I just need to integrate tangent squared which I can't seem to figure how to do. I tried using the trig identity (secx)^2 - 1 but I don't know what to do after.
 

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  • #2
That is a good move.

Hint:
What do you get if you differentiate tan(z)?
 
  • #3
I get (sec(z))^2)
 
  • #4
That is to integrate tan, I said differentiate it!
 
  • #5
Yes, sorry I realized I had made a mistake so when I differentiate tan, I get sec^2 ...
 
  • #6
Indeed, so therefore you DO know an anti-derivative for sec^2, don't you?

Therefore, you should be able to find an anti-derivative for tan^2(z)=sec^2(z)-1
 
  • #7
So simple... thanks a lot
 

What is trig substitution and how is it used to evaluate integrals?

Trig substitution is a technique used to solve integrals that involve expressions containing trigonometric functions. It involves substituting a trigonometric function for a variable in the integral, which can then be simplified and solved using trigonometric identities.

When is trig substitution the best method to use for evaluating an integral?

Trig substitution is most useful when the integral contains a square root of a quadratic expression, or when the integral contains a sum or difference of squares. It is also beneficial when the integral contains a product of trigonometric functions.

What are the common trigonometric substitutions used in evaluating integrals?

The most commonly used trigonometric substitutions are:

  • Sine substitution: used when the integral contains expressions of the form √(a2 - x2)
  • Cosine substitution: used when the integral contains expressions of the form √(a2 + x2)
  • Tangent substitution: used when the integral contains expressions of the form a2 + x2

What are some tips for effectively using trig substitution to evaluate integrals?

Some tips for using trig substitution include:

  • Identify the appropriate trigonometric substitution based on the form of the integral
  • Be familiar with basic trigonometric identities to simplify the integral
  • Check the bounds of integration and adjust them if necessary
  • Be careful with signs and remember to include absolute values when necessary

Are there any limitations or drawbacks to using trig substitution for evaluating integrals?

While trig substitution can be a powerful tool for solving integrals, it is not always the most efficient method. It may also be difficult to identify the appropriate substitution, and the process can become complex for more complicated integrals. Additionally, trig substitution may not work for all types of integrals and may not always produce a closed-form solution.

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