Integrate (ln(x))^2 - Steps and Solution

In summary, the integration of (lnx)^2 involves using the integration by parts formula, setting u=(lnx)^2 and dv=dx. Simplifying the resulting equation, we get x(lnx)^2 - 2∫lnx dx. To integrate lnx, we use integration by parts again, setting u=lnx and dv=dx. This leads to the final solution of xln(x) - x.
  • #1
Rasine
208
0
intergrate (ln(x))^2

so i set u=(lnx)^2...which makes du=2lnx(1/x)

then i set dv=dx...which makes v=x

according to the formula for integration by parts i have

x(lnx)^2- integral x(2lnx)(1/x)
simplifying it i get x(ln)^2-2intergral lnx


and here is where i am stuck...what i the integral of lnx?
 
Physics news on Phys.org
  • #2
The derivative of (lnx)^2=(2lnx)/x.

[edit: of course that's you've written... i glanced and say ln(1/x)... sorry :blushing: ]

A hint for integrating lnx; use parts, taking dv=dx and u=lnx
 
Last edited:
  • #3
How about integration-by-parts once again? :)
 
  • #4
xln(x) - x looks good from where I'm standing.

I just wondered "what function gives ln(x) when differentiated? Well ln(x)' = 1/x. So what if I try xln(x)? Now I get ln(x) + 1. So I need to add something to the mix that gives -1 when differentiated." Hence xln(x) - x.
 
  • #5
ohhh yes...do integration by part again...

thank you!
 

FAQ: Integrate (ln(x))^2 - Steps and Solution

What is the purpose of integrating (ln(x))^2?

The purpose of integrating (ln(x))^2 is to find the antiderivative or the inverse operation of differentiation. This will help us find the original function from its derivative.

What are the steps to integrate (ln(x))^2?

The steps to integrate (ln(x))^2 are:

  1. Use the power rule to rewrite (ln(x))^2 as (ln(x))^2 = ln(x) * ln(x)
  2. Use the product rule to split the product into two separate integrals:∫(ln(x))^2 dx = ∫ln(x) * ln(x) dx = ∫ln(x) dx * ∫ln(x) dx
  3. Integrate each individual integral using the power rule:∫ln(x) dx = x * ln(x) - x + C
  4. Combine the two integrals back together:∫(ln(x))^2 dx = (x * ln(x) - x + C) * (x * ln(x) - x + C) = (x * ln(x))^2 - 2x * ln(x) + x^2 + C

What is the solution to the integral of (ln(x))^2?

The solution to the integral of (ln(x))^2 is (x * ln(x))^2 - 2x * ln(x) + x^2 + C.

Can (ln(x))^2 be integrated using other methods?

Yes, (ln(x))^2 can also be integrated using substitution, where u = ln(x) and du = 1/x dx.

What are some applications of integrating (ln(x))^2?

Integrating (ln(x))^2 can be used in various mathematical and scientific fields such as economics, physics, and engineering. It can also be used in solving differential equations and finding the area under certain curves.

Similar threads

Solve the given problem involving integration
Replies
6
Views
353
Integration problem using inscribed rectangles
Replies
14
Views
1K
How Do You Integrate (x^(1/2))/ln(x) Using Integration by Parts?
Replies
3
Views
1K
Proof that the definite integral of 1/ln(x) doesn't exist
Replies
7
Views
8K
Derivative of ln(ln x) Solution | Step-by-Step Guide
Replies
4
Views
4K
What is the Correct Integral for Finding the Area Below y=0 and Above y=lnx?
Replies
5
Views
1K
How Do You Solve the Integral of 5sin(lnx)?
Replies
4
Views
3K
Solving the Integral ∫dx/(1-x)
Replies
3
Views
1K
Integration by Parts with sin and ln(x)
Replies
10
Views
5K
Integrals: Evaluating ∫ x^{m}* (ln(x))^{2} using Integration by Parts
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top