# Int( X/(x+2)^(1/4)) substitutions

• steve092
In summary, the conversation is about a problem with U-Substitution in finding the integral of X/(x+2)^(1/4). The person attempted to solve it by substituting u=x+2 and du=dx, but got stuck at Int( (u-2)/(u^(1/4)). They ask for help and are then provided with the correct solution of (4/7)(x+2)^(7/4) - (8/3)(x+2)^(3/4) +C.
steve092

## Homework Statement

I'm having a problem with U-Substitution. I get the simple stuff and the trig substitutions, but this problem has been bugging me.
Int( X/(x+2)^(1/4))

## The Attempt at a Solution

I first tried u=x+2 and then du=dx. I then solved for x, where x= U-2.

That put me at Int( (u-2)/(u^(1/4)). And that's where I don't really know how to go on.

You can split (u-2)/u1/4 as u/u1/4-2/u1/4.

So split it then integrate it from there?

Thanks for the help. I got:
(4/7)(x+2)^(7/4) - (8/3)(x+2)^(3/4) +C.

steve092 said:
Thanks for the help. I got:
(4/7)(x+2)^(7/4) - (8/3)(x+2)^(3/4) +C.

That should be correct.

## 1. What is the purpose of using substitutions in the expression Int( X/(x+2)^(1/4))?

Substitutions are used in integrals to simplify the expression and make it easier to solve. In this particular expression, the substitution helps to eliminate the fractional exponent, making it easier to integrate.

## 2. How do I choose the appropriate substitution for this integral?

The choice of substitution depends on the structure of the integral. In this case, we need to choose a substitution that will eliminate the fractional exponent. A common approach is to let u = (x+2)^(1/4), then we can replace x with u^4 in the original expression.

## 3. Can I use more than one substitution in an integral?

Yes, it is possible to use multiple substitutions in an integral. However, it is important to ensure that the substitutions are compatible with each other and do not introduce any new complexities to the integral.

## 4. What is the benefit of using substitutions in an integral?

Substitutions help to simplify integrals and make them easier to solve. They can also help to reveal patterns and make connections between different types of integrals. Additionally, substitutions can also be used to transform integrals into a standard form, making it easier to apply known integration techniques.

## 5. Are there any limitations to using substitutions in integrals?

Substitutions may not always work for every type of integral. In some cases, they may introduce complexities or result in an unsolvable integral. It is important to carefully consider the structure of the integral before choosing a substitution. Additionally, some integrals may require multiple substitutions or other techniques in combination with substitutions to be solved effectively.

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