Efficient Integration of x^3 / (x+1)^10: Tips & Tricks"

Thread starter O.J.
Start date Feb 27, 2007
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Integration

In summary, the most efficient way to integrate x^3 / (x+1)^10 is to use the substitution method, where u = x+1 and du = dx. Other tips for dealing with the high power of (x+1)^10 include using the binomial expansion formula and breaking up the integral into smaller parts. It is also possible to solve the integral without using the substitution method, such as using partial fractions. While the substitution method is usually the most efficient, other techniques such as trigonometric substitutions or integration by parts may also be used. When choosing the most efficient method for integrating x^3 / (x+1)^10, factors such as the complexity of the integral and the individual's familiarity with different

Feb 27, 2007

O.J.

1. Integrate x^3 / (x+1)^10

Tried substitution with u = x+1, or x+1 cubed, but no go

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Feb 27, 2007

cristo

Staff Emeritus

Science Advisor

You could just write it as [tex]x^3(x+1)^{-10}[/tex] and integrate by parts a few times.

Feb 27, 2007

neutrino

O.J. said:

Tried substitution with u = x+1, or x+1 cubed, but no go

If u = x+1, then x = u-1, isn't it?

FAQ: Efficient Integration of x^3 / (x+1)^10: Tips & Tricks"

What is the most efficient way to integrate x^3 / (x+1)^10?

The most efficient way to integrate x^3 / (x+1)^10 is to use the substitution method. Let u = x+1 and du = dx. Then the integral becomes ∫(u-1)^3 / u^10 * du. This can be simplified using the binomial expansion formula, making it easier to integrate.

What are some tips for dealing with the high power of (x+1)^10?

One tip is to use the binomial expansion formula to simplify the integral. Another tip is to break up the integral into smaller parts, such as integrating (x+1)^3 / (x+1)^10 and (x+1)^7 / (x+1)^10 separately.

Can the integral of x^3 / (x+1)^10 be solved without using the substitution method?

Yes, it is possible to solve the integral without using the substitution method. One method is to use partial fractions, where the integrand is broken down into simpler fractions that can be integrated separately.

Is there a faster way to integrate x^3 / (x+1)^10 compared to the substitution method?

The substitution method is usually the most efficient method for integrating x^3 / (x+1)^10. However, some other techniques that can be used include using trigonometric substitutions or using integration by parts.

What should be considered when choosing the most efficient method for integrating x^3 / (x+1)^10?

The most efficient method may vary depending on the specific integral and the individual's familiarity with different integration techniques. Factors to consider may include the complexity of the integral, the availability of certain techniques, and the individual's preferred method of integration.