How do you integrate (ln(x))^2? dx

• maxfails
In summary, there seems to be some confusion about the property ln x^n = n ln x and the use of integration by parts with 1 as the function. There is also a difference between (ln(x))^2 and ln(x^2).
maxfails
it seems you can't use the property ln x^n = n ln x.

I'm thinking there's integration by parts involved but not sure.

maxfails said:
I'm thinking there's integration by parts involved but not sure.

Hi maxfails!

Yes, use integration by parts with 1 as the function.

ln xn = t
x = et
dx = et dt

so initial eqn becomes

$$\int t^n e^t dt$$

and now integrate by parts

ln xn = t
x = et
dx = et dt

so initial eqn becomes

$$\int t^n e^t dt$$

$$ln \ x^n = t$$

$$e^{ln \ x^n} = e^t$$

$$x^n=e^t$$

$$\frac{d}{dt} \ (x^n)=\frac{d}{dt} \ (e^t)$$

$$0=e^t$$

remember that:

$$x=exp \ y \Leftrightarrow y=ln \ x$$

So

$$0=e^t \Leftrightarrow t = ln \ 0$$

Since ln 0 is undefined, so t is undefined too...

maxfails said:
it seems you can't use the property ln x^n = n ln x.

The problem is that this formula is $\ln(x^n)=n\ln(x)$, but you are now interested in $(\ln(x))^n$ which is different. You probably knew this, but didn't sound very sure about it.

I'm thinking there's integration by parts involved but not sure.

Well tiny-tim of course answered quite sufficiently already, but I thought I would like to say that personally I like writing recursive formulas such as this:

$$(\ln(x))^n = D_x \big(x(\ln(x))^n\big) - n(\ln(x))^{n-1}$$

fundoo, optics.tech, those were quite confusing comments

Umm there's a difference between (ln(x))^2 and ln(x^2). The first is what you seem to have, the latter is 2*ln(x).

1. What is the general formula for integrating (ln(x))^2?

The general formula for integrating (ln(x))^2 is ∫ (ln(x))^2 dx = x(ln(x))^2 - 2xln(x) + 2x + C.

2. Can (ln(x))^2 be simplified before integrating?

Yes, (ln(x))^2 can be simplified to ln(x^2) before integrating.

3. How do you integrate (ln(x))^2 using substitution?

To integrate (ln(x))^2 using substitution, let u = ln(x), then du = 1/x dx. The integral becomes ∫ u^2 du, which can be easily solved using the power rule.

4. Is there another method for integrating (ln(x))^2?

Yes, you can also integrate (ln(x))^2 by parts. Let u = (ln(x))^2 and dv = dx, then du = 2ln(x)/x dx and v = x. The integral becomes ∫ (ln(x))^2 dx = x(ln(x))^2 - 2∫ xln(x)/x dx. The second term can be simplified to -2∫ ln(x) dx, which can then be solved using substitution.

5. Can (ln(x))^2 be integrated using a calculator?

Yes, most scientific calculators have a built-in function for integrating (ln(x))^2. Simply input the function and the limits of integration to get the result.

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