Derive integral of sqrt(x^2-a^2)

Thread starter coverband
Start date Dec 5, 2008
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Derive Integral

In summary, the formula for the integral of sqrt(x^2-a^2) is ∫ √(x^2-a^2) dx = (x/2)√(x^2-a^2) + (a^2/2)ln|x+√(x^2-a^2)| + C. It can be derived using the substitution method by letting x = asec(u), then dx = asec(u)tan(u) du. The purpose of deriving this integral is to calculate the area under the curve of the function √(x^2-a^2). It can be simplified further using trigonometric identities and algebraic manipulation to (x/2)√(

Dec 5, 2008

coverband

Sorry me again

THE CRC tables define the above integral as

using sqrt(x^2-a^2)

let t(x) = sqrt(x^2 - a^2)

integral = 1/2( x T(x) - a^2 log( x + T(x))

HOW do they arrive at this. Thanks

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Dec 5, 2008

morphism

Science Advisor

Homework Helper

Are t and T the same?

And have you tried using a trig or hyperbolic substitution?

1. What is the formula for the integral of sqrt(x^2-a^2)?

The formula for the integral of sqrt(x^2-a^2) is ∫ √(x^2-a^2) dx = (x/2)√(x^2-a^2) + (a^2/2)ln|x+√(x^2-a^2)| + C

2. How do you derive the integral of sqrt(x^2-a^2)?

The integral of sqrt(x^2-a^2) can be derived by using the substitution method. Let x = asec(u), then dx = asec(u)tan(u) du. Substituting these values into the integral and simplifying will result in the formula mentioned in the first question.

3. What is the purpose of deriving the integral of sqrt(x^2-a^2)?

The purpose of deriving the integral of sqrt(x^2-a^2) is to be able to calculate the area under the curve of the function √(x^2-a^2), which is useful in many applications in science and engineering.

4. Can the integral of sqrt(x^2-a^2) be simplified further?

Yes, the integral of sqrt(x^2-a^2) can be simplified further by using trigonometric identities and algebraic manipulation. The simplified form is (x/2)√(x^2-a^2) + (a^2/2)arcsin(x/a) + C.

5. Are there any other methods to derive the integral of sqrt(x^2-a^2)?

Yes, there are other methods such as using the integration by parts method or using the trigonometric substitution method. However, the substitution method is the most commonly used and straightforward method to derive the integral of sqrt(x^2-a^2).