Is the Integral of ln(1+2^x) Correctly Calculated?

  • Thread starter UrbanXrisis
  • Start date
  • Tags
    Integration
In summary, the integral of ln(1+2^x) can be solved by using substitution and integration by parts. However, the result obtained by differentiating the original function appears to be the same as the integral, which means there may be a mistake in the process.
  • #1
UrbanXrisis
1,196
1
what is the integral of [tex]ln(1+2^x)[/tex]

[tex]\int ln(1+2^x)=\frac{2^xln(2)}{1+2^x} [/tex]

is this correct?
 
Physics news on Phys.org
  • #2
Differentiate and see what you get.
 
  • #3
Try

[tex] u = 2^x+1, du = 2^xln(2) [/tex]

Then

[tex] \int ln(1+2^x) dx = ln(2)\int (u-1)ln(u) du [/tex]

Which can be done by parts.
 
  • #4
UrbanXrisis said:
what is the integral of [tex]ln(1+2^x)[/tex]

[tex]\int ln(1+2^x)=\frac{2^xln(2)}{1+2^x} [/tex]

is this correct?

That result you got looks suspiciously like the derivative of [itex] \ln(1+2^x) [/itex]! What I'm saying is, it looks like you differentiated using the chain rule:

let u = 1 + 2^x

[tex] \frac{d}{dx}[\ln(1+2^x)] = \frac{d}{dx}(\ln u) [/tex]

[tex] = \frac{1}{u} \frac{du}{dx} [/tex]

[tex] = \left(\frac{1}{1+2^x}\right) \frac{d}{dx}(1+2^x) [/tex]

[tex] = \frac{(\ln2)2^x}{1+2^x} [/tex]

But you were supposed to integrate! :smile:

I just thought I'd point that out, so you could see the mistake.
 

Related to Is the Integral of ln(1+2^x) Correctly Calculated?

1. What is integration?

Integration is the process of combining different parts or components into a whole. In the scientific context, it often refers to the process of incorporating new knowledge or techniques into existing theories or systems.

2. Why is integration important?

Integration allows for a more comprehensive understanding of complex systems and phenomena. By combining different perspectives and approaches, integration can lead to new insights and advancements in scientific research.

3. How do I know when to start integrating?

The need for integration often arises when there is a gap or inconsistency in existing knowledge or theories. It can also be beneficial to start integrating when there are conflicting findings or when a new discovery challenges previous assumptions.

4. What are some strategies for starting the integration process?

Some strategies for starting the integration process include identifying common themes or patterns among different theories or research findings, collaborating with experts from different disciplines, and conducting a thorough literature review.

5. How can integration be effectively implemented?

Effective integration requires open-mindedness, communication, and collaboration among scientists from different fields. It is also important to critically evaluate the evidence and to consider potential limitations or biases in the integration process.

Similar threads

Replies
7
Views
1K
  • Introductory Physics Homework Help
Replies
10
Views
310
  • Introductory Physics Homework Help
Replies
1
Views
1K
  • Introductory Physics Homework Help
Replies
3
Views
185
  • Introductory Physics Homework Help
2
Replies
64
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
329
  • Introductory Physics Homework Help
Replies
11
Views
106
  • Introductory Physics Homework Help
Replies
28
Views
542
  • Introductory Physics Homework Help
Replies
12
Views
697
Replies
0
Views
290
Back
Top