How can the reduction formula be used to find the integral of tan^4 x?

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In summary, the conversation discusses a reduction formula for solving integrals involving tan functions, with a hint to use the formula twice to find the integral of tan^4 x. The approach is to use methods of integration to prove that the left side equals the right side.
  • #1
silicon_hobo
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Homework Statement


a)Prove the reduction formula:

[tex]\int\ tan^n\ x\ dx\ =\ \frac{1}{n-1}tan^{n-1}\ x\ -\int\ tan^{n-2}\ x\ dx[/tex]

Hint: first write [tex]tan^n x[/tex] as [tex]tan^{n-2} \ x\ tan^2\ x[/tex] and the rewrite using [tex]tan^2\ x+1=sec^2\ x[/tex].

b) Use the formula twice to find [tex]\int\ tan^4\ dx[/tex]

The Attempt at a Solution



I not sure what they're asking for when they say "prove". How should I begin with this one? Thanks
 
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  • #2
Use methods of integration to show the left side equals the right. From the form of the right side, it should be pretty obvious which method to use.
 

1. What is the Reduction Formula?

The Reduction Formula is a mathematical concept used to evaluate integrals involving products of powers of trigonometric functions. It is a method for simplifying complex integrals into a series of simpler integrals.

2. How does the Reduction Formula work?

The Reduction Formula works by repeatedly applying integration by parts to a given integral, until it can be expressed in terms of a simpler integral. This process is continued until the integral is reduced to a form that can be evaluated using basic integration rules.

3. What are the benefits of using the Reduction Formula?

The Reduction Formula allows for the evaluation of integrals that would otherwise be very difficult or impossible to solve. It also helps to simplify complicated integrals into a series of simpler integrals, making them easier to solve.

4. What are the limitations of the Reduction Formula?

The Reduction Formula is not applicable to all types of integrals. It is specifically designed for integrals involving products of powers of trigonometric functions. It also requires a good understanding of integration by parts and basic integration rules.

5. Can the Reduction Formula be used in other areas of mathematics?

Yes, the Reduction Formula can be applied in other areas of mathematics, such as differential equations, by converting them into integrals and then using the Reduction Formula to solve them. It can also be used in physics and engineering to solve complex problems involving integrals.

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