# How Do You Simplify Complex Root Integrals?

• transgalactic
In summary, the conversation discusses the substitution method for solving the given integral. The participants also mention the creation of additional roots and suggest using fractional powers to simplify the process.
transgalactic
i tried this:
$$\int \frac{1-\sqrt{x+1}}{1+\sqrt[3]{x+1}}=\int \frac{1-\sqrt{x+1}}{1+\sqrt[3]{x+1}}*\frac{1+\sqrt{x+1}}{1+\sqrt{x+1}}*\frac{1-\sqrt[3]{x+1}+(x+1)^{\frac{3}{2}}}{1-\sqrt[3]{x+1}+(x+1)^{\frac{3}{2}}}$$
but when i got read of 2 roots i got another two roots which are more complicated
??

transgalactic said:
$$\int \frac{1-\sqrt{x+1}}{1+\sqrt[3]{x+1}}$$

Hi transgalactic!

Hint: substitute!

i tried t=(x+1)^1/3
but it creates anoted roots in the dt
??

transgalactic said:
i tried t=(x+1)^1/3
but it creates anoted roots in the dt
??

uhh?

dx = … ?

anyway, (x+1)1/6 might be easier.

but i don't have members of 1/6 power
??

Sure you do. a^(1/2) = a^(1/6)^3, and b^(1/3) = b^(1/6)^2.

## 1. What is a root integral?

A root integral is an integral that involves finding the inverse function of a given function. It is typically written in the form of ∫f(x)^(1/n) dx, where n is the root number.

## 2. What is the general approach to solving a root integral?

The general approach to solving a root integral is to first rewrite the integral in terms of an inverse trigonometric function or a substitution that will allow for the use of the power rule. Then, use integration by parts or other integration techniques to evaluate the integral.

## 3. What are some common mistakes to avoid when solving a root integral?

Some common mistakes to avoid when solving a root integral include forgetting to use the power rule, not properly applying substitution or integration by parts, and incorrectly evaluating the integral limits.

## 4. Can all root integrals be solved analytically?

No, not all root integrals can be solved analytically. Some integrals may require the use of numerical methods or cannot be solved at all.

## 5. Are there any tips for solving root integrals more efficiently?

One tip for solving root integrals more efficiently is to first check if the integral can be rewritten in terms of a simpler function, such as a logarithm or exponential function. Additionally, practicing various integration techniques and knowing when to use them can help with efficiency.

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