# Integrate 1/x(2/3) - Solve for 3 Cube Root 3

• Roodles01
In summary, the conversation discusses the use of integration by parts and the difficulty in solving for 1/(x^(2/3)). The standard formulas for 1/x and xn are mentioned, but they do not solve the problem. The solution is found by putting n=(-2/3) into the formula x^(n+1)/(n+1). The importance of practice and asking questions is emphasized.
Roodles01
knowing the standard form for integration by parts is
∫ f(x)g'(x) dx = f(x)g(x) - ∫f'(x)g(x) dx

I have what is an innocuous looking part of an equation which I can't solve.

the f(x) part in this case is;
ln(5x) which is easy enough i.e. 1/x

the second part 1/(x(2/3)) is the bit I can't solve.

The standard I have for
1/x is ln(x)+c
& the standard I have for xn is (1/(n+1))xn+1+c

But these don't solve this for me

I have checked on WolframAlpha & NumberEmpire & they give the same answer
3 cuberoot 3

I have tried just this bit by itself & go t nowhere. Could someone help with how I should get 3 cuberoot 3, please.

1/x^(2/3)=x^(-2/3). So put n=(-2/3) into x^(n+1)/(n+1).

I'm being thick here, but doesn't (n+1)/(n+1) = 1

So x1 = x ?

Sorry.

Last edited:
Roodles01 said:
I'm being thick here, but doesn't (n+1)/(n+1) = 1

So x1 = x ?

Sorry.

Ok, I'll be a little clearer. I meant the formula you referred to $\frac{x^{n+1}}{n+1}$.

Yes, it is true that x1= x! However, the anti-derivative of x1 is x1+1/(1+1)= x2/2.

I am surprised that you are being asked to use "integration by parts" but do not know how to integrate xn.

I shall try to be more numerically erudite in future!

I think that sometimes I have to ask stupid questions when I have come to the end of my tether & I can't see the wood for the trees.

Practice makes perfect & asking stupid questions should embarrass me into remembering it properly.

Prepare for more along the same lines in the future.

## 1. What is the process for integrating 1/x?

The process for integrating 1/x is to use the natural logarithm function. The integral of 1/x is ln|x| + C, where C is the constant of integration.

## 2. How do I solve for 3 cube root 3?

To solve for 3 cube root 3, you can use the power rule for radicals. This states that the nth root of x to the power of n is equal to x. In this case, 3 cube root 3 is equal to 3.

## 3. What is the value of the constant of integration in the integral of 1/x?

The value of the constant of integration in the integral of 1/x is arbitrary and can be represented by any letter, typically C. It is added to the solution to account for all possible solutions.

## 4. Can I simplify the integral 1/x(2/3)?

Yes, you can simplify the integral 1/x(2/3) by factoring out the constant 2/3. This will result in (2/3)*ln|x| + C.

## 5. Is there a general formula for solving integrals?

There is no single general formula for solving integrals. The method used to solve an integral depends on the form of the integral, such as whether it is a polynomial, trigonometric function, or exponential function. Different techniques, such as substitution and integration by parts, are used to solve different types of integrals.

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