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.