# Integral of 1/(sqrt(x^2 *ln(x)) from e to e^2 No idea.

• Lo.Lee.Ta.
In summary, the integral of 1/(sqrt(x^2 *ln(x)) from e to e^2 can be simplified by changing the variables to ln(x) = y. After integration, the final answer is 2sqrt(2) - 2.
Lo.Lee.Ta.
Integral of 1/(sqrt(x^2 *ln(x)) from e to e^2... No idea. :(

1. ∫e to e^2 of (1/(√x2ln(x)))dx

2. So confused! :(

Okay... So I tried something!

∫e to e^2 of [-2/3(x2*ln(x))-3/2 * 1/(x + 2xln(x))] |e to e^2

I got the 1/(x+2xln(x)) because I was trying to think of some way to cancel out the
x+2xln(x) that results from the chain rule...

Don't think this is right... If not, what am I supposed to do here?

Lo.Lee.Ta. said:
1. ∫e to e^2 of (1/(√x2ln(x)))dx

2. So confused! :(

Okay... So I tried something!

∫e to e^2 of [-2/3(x2*ln(x))-3/2 * 1/(x + 2xln(x))] |e to e^2

I got the 1/(x+2xln(x)) because I was trying to think of some way to cancel out the
x+2xln(x) that results from the chain rule...

Don't think this is right... If not, what am I supposed to do here?

For x > 0 we have ##\sqrt{x^2 \ln(x)} = x \sqrt{\ln(x)},## because ##\sqrt{x^2} = x## for x > 0. Change variables to ln(x) = y.

Oh. I think I know what you mean. It's not as hard as I thought it would be!

∫e to e^2 of [1/(xsqrt(ln(x)))]

u = ln(x)
du= (1/x)dx

∫e to e^2 [1/(sqrt(u))]du

= ∫ u^(-1/2) du = 2(u)^1/2 |e to e^2

= 2(ln(e^2))^1/2 - 2(ln(e))^1/2

= 2(2)^1/2 - 2(1)^1/2

= 2sqrt(2) - 2 <----Should be answer!

Thanks, Ray Vickson! :D

## 1. What is the integral of 1/(sqrt(x^2 *ln(x)) from e to e^2?

The integral of 1/(sqrt(x^2 *ln(x)) from e to e^2 is equal to 2 - 2ln(2).

## 2. How do I solve the integral of 1/(sqrt(x^2 *ln(x)) from e to e^2?

To solve this integral, you can use the substitution method by letting u = ln(x). This will result in the integral becoming 1/sqrt(u^2) from 1 to 2. Then, you can use the trigonometric substitution method with u = tan(theta) to further simplify the integral.

## 3. Can this integral be evaluated using any other method?

Yes, you can also use integration by parts with u = 1 and dv = 1/(sqrt(x^2 *ln(x)) to solve the integral.

## 4. What is the significance of the limits e and e^2 in this integral?

The limits e and e^2 represent the natural logarithmic values of 1 and 2, respectively. This is because the function 1/(sqrt(x^2 *ln(x)) is undefined at x = 1 and x = 2, but it approaches 0 as x approaches these values.

## 5. Can this integral be solved using a calculator or numerical methods?

Yes, you can use a calculator or numerical methods to approximate the value of the integral. However, the exact solution can only be obtained through algebraic manipulation and integration techniques.

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