Integration by Reduction Formulae

Thread starter saladfinger16
Start date Dec 15, 2010
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Formulae Integration Reduction

In summary, Integration by Reduction Formulae is a mathematical technique used to solve integrals of functions that cannot be integrated directly. It is typically used when the integrand contains a polynomial expression, trigonometric functions, or a combination of both. To apply this technique, the integrand is simplified using algebraic or trigonometric identities, and a known integration formula is used to derive a new formula that can be used to solve the integral. This process may need to be repeated multiple times until the integral can be solved. The benefits of using Integration by Reduction Formulae include the ability to evaluate integrals that cannot be solved using basic techniques, simplifying complicated integrals, and finding the antiderivative of a function. However, it is

Dec 15, 2010

saladfinger16

Can anyone help me out with a proof for the integral that's in the attached images, its driving me nuts

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Dec 15, 2010

snipez90

You could try the trig substitution x = a*tan(u).

Or note that

[tex]\int\frac{dx}{(x^2 + a^2)^n} = \int\frac{dx}{(x^2 + a^2)^{n-1}} - \int\frac{x^2 + a^2 - 1}{(x^2 + a^2)^n}\,dx[/tex]

and try to work out the last integral on the right.

1. What is Integration by Reduction Formulae?

Integration by Reduction Formulae is a mathematical technique used to solve integrals of functions that cannot be integrated directly. It involves using a known integration formula to derive a new formula that can be used to solve more complex integrals.

2. When is Integration by Reduction Formulae used?

Integration by Reduction Formulae is typically used when the integrand (the function being integrated) contains a polynomial expression, trigonometric functions, or a combination of both. It is also useful when solving integrals involving rational functions and logarithmic functions.

3. How is Integration by Reduction Formulae applied?

To apply Integration by Reduction Formulae, the integrand is first simplified using algebraic or trigonometric identities. Then, a known integration formula is used to derive a new formula that can be used to solve the integral. This process may need to be repeated multiple times until the integral can be solved.

4. What are the benefits of using Integration by Reduction Formulae?

Integration by Reduction Formulae allows for the evaluation of integrals that cannot be solved using basic integration techniques. It also helps simplify complicated integrals, making them easier to solve. Additionally, it can be used to find the antiderivative of a function, which is useful in many areas of mathematics and science.

5. Are there any limitations to Integration by Reduction Formulae?

Integration by Reduction Formulae can only be applied to certain types of integrals, specifically those that contain polynomial or trigonometric functions. It also requires knowledge of known integration formulas and algebraic/trigonometric identities. This technique may not always yield a closed-form solution, and in some cases, numerical methods may need to be used instead.