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

[UrbanXrisis]

what is the integral of ln(1+2^x)

\int ln(1+2^x)=\frac{2^xln(2)}{1+2^x}

is this correct?

 

[whozum]

Differentiate and see what you get.

 

[whozum]

Try

u = 2^x+1, du = 2^xln(2)

Then

\int ln(1+2^x) dx = ln(2)\int (u-1)ln(u) du

Which can be done by parts.

 

[cepheid]

what is the integral of ln(1+2^x)

\int ln(1+2^x)=\frac{2^xln(2)}{1+2^x}

is this correct?

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

let u = 1 + 2^x

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

= \frac{1}{u} \frac{du}{dx}

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

= \frac{(\ln2)2^x}{1+2^x}

But you were supposed to integrate!

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