Find integral of fraction

In summary: If so, the substitution ##t = \frac{1}{2} \sec^{-1} (x+1)## might be a good start, leading to a simple form.In summary, the conversation discusses finding the integration of a complex function involving a square root and multiple trigonometric substitutions. However, the conversation suggests using the substitution ##\tan^{-1}f(x)## as a possible solution. One person also suggests using the substitution ##t = \frac{1}{2} \sec^{-1} (x+1)## which may lead to a simpler form.
  • #1
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Homework Statement
Find the integration of .... (for complete question see "attempt at solution")
Relevant Equations
Integration
Find integration of:
[itex] \frac {1}{(x+1)(x^2 + x -1)^\frac{1}{2}} [/itex]

What I did:
1. Use completing square method for the term inside the square root

2. Use trigonometry substitution (I use secan)

3. After simplifying, use another trigonometry substitution (I use weierstrass substitution)

4. Use another trigonometry substitution (I use tan)My working was so long and taking too much time. Is there any other way to solve this integration?

Thanks
 
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  • #4
Can you bring it into the form ## \dfrac{f'(x)}{1+f(x)^2}## where ##f(x)## is the quotient of a polynomial in ##x## and the root ##\sqrt{x^2+x-1}## in the denominator? Find ##f(x)## and then you get ##\tan^{-1} f(x)## as antiderivative (up to a sign).
 
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  • #5
I am afraid this integral has no easy route to calculate it...

Unless of course someone whispers you "Psst, this integral is of the form ##\tan^{-1}f(x)## " in which case you could try to find f(x) as suggested in post #4.
 
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  • #6
Delta2 said:
I am afraid this integral has no easy route to calculate it...

Unless of course someone whispers you "Psst, this integral is of the form ##\tan^{-1}f(x)## " in which case you could try to find f(x) as suggested in post #4.

Unfortunately even though I heard the whisper I still can not do anything :doh:

fresh_42 said:
Can you bring it into the form ## \dfrac{f'(x)}{1+f(x)^2}## where ##f(x)## is the quotient of a polynomial in ##x## and the root ##\sqrt{x^2+x-1}## in the denominator? Find ##f(x)## and then you get ##\tan^{-1} f(x)## as antiderivative (up to a sign).

Sorry I can not. You mean I have to find f(x), which is the quotient of a certain polynomial (let say g(x)) divided by x2 + x - 1?

g(x) : (x2 + x - 1) and the quotient of this long division is f(x)? Before finding f(x), I need to find g(x) first?

Thanks
 
  • #7
Well, WolframAlpha has the solution: something like ##-\arctan \dfrac{x+a}{b\sqrt{x^2+x-1}}##, so the reverse inspection, i.e. the derivative of ##\arctan## points the way.

I thought about the Weierstraß substitution ##t=\tan \frac{x}{2}## which is normally used to turn trigonometric integrands into polynomials, and apply it the other way around, but it didn't look promising. So I have no idea what the best purely forward path is.
 
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  • #8
ok. thank you very much for the help fresh_42 and delta2
 
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  • #9
Considering the substitutions your text as armed you with so far, I suspect that there may have been a misprint in your problem assignment and the integration is actually$$\int \frac {dx} {(2x + 1) \sqrt {x^2 + x -1}}$$
 

1. What is the definition of an integral?

An integral is a mathematical concept that represents the accumulation of a quantity over a given interval. It can also be interpreted as the area under a curve.

2. What is the difference between definite and indefinite integrals?

A definite integral has specific limits of integration, while an indefinite integral does not have any limits and represents a family of functions.

3. How do I find the integral of a fraction?

To find the integral of a fraction, you can use the power rule, where you add 1 to the exponent and divide the coefficient by the new exponent. You can also use substitution or integration by parts depending on the complexity of the fraction.

4. What is the purpose of finding the integral of a fraction?

Finding the integral of a fraction can be used to solve various problems in physics, engineering, and economics. It can also help in finding the total change or net value of a quantity.

5. Are there any limitations to finding the integral of a fraction?

Yes, there are certain types of functions that cannot be integrated using elementary functions and require more advanced techniques. Additionally, the integral of a fraction may not always exist if the function is not continuous or if the limits of integration are infinite.

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