Integral with roots on bottom and top

In summary, the best approach to solving this integral is to use a substitution of x^1/6, as it allows for the elimination of both powers of x in a manageable way. One possible method is to set x=u^6, where sqrt(x)=u^3, cubed root(x)=u^2, and dx/du=6u^5. However, this approach has not been verified and the given answer contains several powers of u, so further steps may be required.
  • #1
rasen58
71
2
It's the integral of sqrt(x)/(cubed root(x) + 1)
I tried regular u substitution but that didn't let me get rid of all the x's.
I also just tried long division but that gave me an answer that didn't match with the actual answer to the problem.

The actual answer is 6[1/7 x^(7/6) - 1/5 x^(5/6) + 1/3 x^(1/6) - x^(1/6) + arctan(x^(1/6)) ] + C
How do I go about doing this?
 
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  • #2
The roots suggest a substitution of x1/6 as this allows to get rid of both powers of x in a reasonable way.
I did not check if it works, however. The answer is full of powers of that, so it looks good.
 
  • #3
try setting x=u6. The sqrt(x) = u3, cubed root(x) = u2 and dx/du = 6u5. The rest is left to the student...
 

What is an integral with roots on bottom and top?

An integral with roots on bottom and top is a type of integral in which the function being integrated has both a numerator and denominator that contain roots.

How do you solve an integral with roots on bottom and top?

To solve an integral with roots on bottom and top, you can use the substitution method, where you substitute the root expression with a new variable. This will allow you to simplify the integral and solve it using traditional integration techniques.

What is the significance of having roots on bottom and top in an integral?

The presence of roots on both the numerator and denominator in an integral indicates that the function may have a singularity at that point. This means that the function may not be defined or may be approaching infinity at that point, which can affect the overall value of the integral.

Are there any special techniques for evaluating an integral with roots on bottom and top?

Yes, there are a few special techniques that can be used for evaluating integrals with roots on bottom and top. These include the substitution method, partial fractions, and integration by parts.

Can an integral with roots on bottom and top be written as a simpler form?

Yes, an integral with roots on bottom and top can be simplified using algebraic or trigonometric identities. This can help to make the integral easier to solve and evaluate.

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