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What is $ \displaystyle\int_0^1 \frac{\ln(1+bx)}{1+x} dx $?

I call it $f(b)$ and differentiate with respect to be $b,$ a bit of partial fractions and the $x$ integral can be done. Then I integrate with respect to $b$ and get a bit lost.

Can some of the terms be expressed in terms of dilogarithms? I get lost in the details! Could we avoid all this just go straight to dilogarithms (with a cunning substitution)?

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Check out this question where $b = -1$. –  user2468 Mar 8 '12 at 16:09
    
Well, if you substitute $bx = b - bu - 1$, you will get $b \int \frac{\ln (1-u)}{bu + (2b-1)} dx + b \int \frac{\ln b}{bu + (2b-1)}$. Not sure how to integrate the first one though. –  user2468 Mar 8 '12 at 16:16
    
Can you show your working? Seems like differentiating wrt $b$ might work. –  Aryabhata Mar 8 '12 at 16:33
    
Maple says$-dilog \biggl(\frac{b}{b - 1}\biggr) + dilog \biggl(\frac{2 b}{b - 1}\biggr) + \operatorname{ln} (1 + b) \operatorname{ln} \biggl(\frac{2 b}{b - 1}\biggr)$ –  GEdgar Mar 8 '12 at 20:47
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4 Answers

Let $x = \frac{1}{b} \frac{1-u}{u}$. The integration range $0\leqslant x \leqslant 1$ translates to $\frac{1}{1+b} \leqslant u \leqslant 1$, for $b>0$ $$ \frac{\log(1+ b x)}{1+x}\mathrm{d} x \to \frac{-\log(u)}{1 - (1-b) u} \frac{\mathrm{d} u}{u} = \frac{\log(u)}{u} \left(1 + \frac{(1-b) u}{1-(1-b) u} \right) $$ Now $$ \begin{eqnarray} \int \frac{\log(u)}{1 - (1-b) u} \frac{\mathrm{d} u}{u} &\stackrel{\color\red{\text{by parts}}}{=}& \frac{\log^2(u)}{2} - \log(u) \cdot \log(1-(1-b)u) - \int \frac{\log(1-(1-b)u)}{u} \mathrm{d} u \\ &=& \frac{\log^2(u)}{2} - \log(u) \cdot \log(1-(1-b)u) - \operatorname{Li_2}\left((1-b) u\right) + C \end{eqnarray} $$ where a definition of dilogarithm was used $\operatorname{Li_2}^\prime(z) = -\frac{\log(1-z)}{z}$. Now assuming $b$ is such that the anti-derivatives is smooth: $$ \begin{eqnarray} \int_0^1 \frac{\log(1+b x)}{1+x} \mathrm{d} x &=& \int\limits_{\frac{1}{1+b}}^1 \frac{-\log(u)}{1 - (1-b) u} \frac{\mathrm{d} u}{u} \\ &=& \operatorname{Li_2}\left(1-b\right) - \operatorname{Li_2}\left(\frac{1-b}{1+b}\right) + \frac{1}{2} \log^2\left(1+b\right) +\log\left(1+b\right) \log\left(\frac{2b}{1+b}\right) \end{eqnarray} $$

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I knew you would solve this one. –  Pedro Tamaroff Apr 20 '12 at 23:01
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$$\Phi(b)=\int\limits_0^1 \frac{\log(1+bx)}{x+1}dx$$

$$\Phi'(b)=\int\limits_0^1 \frac{x}{(bx+1)(x+1)}dx$$

$$\Phi'(b)=\int\limits_0^1 \frac{1}{bx+1}dx-\int\limits_0^1 \frac{1}{(x+1)(bx+1)}dx$$

$$\Phi'(b)=\frac{\log(b+1)}{b}-\frac{1}{1-b}\int\limits_0^1 \left(\frac{1}{x+1}- \frac{b}{bx+1}\right) dx$$

$$\Phi'(b)=\frac{\log(b+1)}{b}-\frac{\log2}{1-b}+\frac{\log(b+1)}{1-b}$$

With a different approach:

$$\Phi(b)=\int\limits_0^1 \frac{\log(1+bx)}{x+1}dx=\int\limits_0^1 \int\limits_0^{bx}\frac{dx \cdot dy}{(1+y)(1+x)}$$

Putting $y=bxu$ gives

$$\Phi(b)=\int\limits_0^1 \frac{\log(1+bx)}{x+1}dx=\int\limits_0^1 \int\limits_0^{1}\frac{b \cdot x dx \cdot du}{(1+bxu)(1+x)}$$

Some partial fraction decomposition and trickery will give you

$$\Phi(b)=\int\limits_0^b \frac{\log(1+x)}{x}dx + \log2 \log|b-1|+\int\limits_0^b\frac{\log(1+x)}{1-x}dx$$

Note that the first integral is a Dilogarithm and the latter is most probably one too (there is a comment saying Maple gives the answer in terms of te Dilog)

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For $(1+bx) > 0, \ln(1+bx) \leq bx$

If not evaluating the exact value, at least we can find that

$$\int_0^1 \frac{\ln(1+bx)}{1+x} dx \leq \int_0^1 \frac{bx}{1+x} dx = b(1-\ln 2)$$

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If $b<1$, one may try:

$$\int_{0}^{1} \frac{\log(1+bx)}{1+x} dx = \int_{0}^{1} \log(1+bx)\cdot\frac{1}{1+x} dx = \int_{0}^{1} \left(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(bx)^{n}}{n} \right)\left(\sum_{m=0}^{\infty} (-1)^{m}x^{m}\right) dx$$

Then:

$$ \int_{0}^{1} \left(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(bx)^{n}}{n} \right)\left(\sum_{m=0}^{\infty} (-1)^{m}x^{m}\right) = \sum_{n=1}^{\infty} \sum_{m=0}^{\infty} \frac{(-1)^{m+n+1}b^{n}}{n} \int_{0}^{1} x^{m+n} dx = \sum_{n=1}^{\infty} \sum_{m=0}^{\infty} \frac{(-1)^{m+n+1}b^{n}}{n(m+n+1)} $$

Gaffe: Or it would, if the integral didn't diverge. (the numerical packages I'm using are both giving nonsensical results, which either means its an improper integral beyond their scope of dealing with improper integrals or that it doesn't converge) Does it converge for any $b$?

I mucked up the power series for the denominator. Fixed that.

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Does it converge for any $b$? It does, as soon as the integrand is well defined on $(0,1)$, that is, for every $b\geqslant-1$. –  Did Mar 8 '12 at 19:23
    
@deoxygerbe Please see my answer on how to arrive at the closed form result in terms of dilogarithms. –  Sasha Apr 21 '12 at 14:37
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