Integrating $\int_0^\infty \frac{\log x}{(1+x)^3}\,\operatorname d\!x$ using residues

Question

I am trying to use residues to compute $$\int_0^\infty\frac{\log x}{(1+x)^3}\,\operatorname d\!x.$$My first attempt involved trying to take a circular contour with the branch cut being the positive real axis, but this ended up cancelling off the term I wanted. I wasn't sure if there was another contour I should use. I also had someone suggest using the substitution $x=e^z$, so the integral becomes $$\int_{-\infty}^\infty\frac{ze^z}{(1+e^z)^3}\,\operatorname d\!z$$so that the poles are the at the odd multiples of $i\pi$. I haven't actually worked this out, but it does not seem like the solution the author was looking for (this question comes from an old preliminary exam).

Any suggestions on how to integrate?

Answer 1

Consider the integral

$$\oint_C dz \frac{\log^2{z}}{(1+z)^3}$$

where $C$ is a keyhole contour in the complex plane, about the positive real axis. This contour integral may be seen to vanish along the outer and inner circular contours about the origin, so the contour integral is simply equal to

$$\int_0^{\infty} dx \frac{\log^2{x}-(\log{x}+i 2 \pi)^2}{(1+x)^3} = -i 4 \pi \int_0^{\infty} dx \frac{\log{x}}{(1+x)^3}+4 \pi^2 \int_0^{\infty} dx \frac{1}{(1+x)^3}$$

By the residue theorem, the contour integral is also equal to $i 2 \pi$ times the residue at the pole $z=-1=e^{i \pi}$. In this case, with the triple pole, we have the residue being equal to

$$\frac12 \left [ \frac{d^2}{dz^2} \log^2{z}\right]_{z=e^{i \pi}} = 1-i \pi$$

Thus we have that

$$-i 4 \pi \int_0^{\infty} dx \frac{\log{x}}{(1+x)^3}+4 \pi^2 \frac12 = i 2 \pi + 2 \pi^2$$

which implies that

$$\int_0^{\infty} dx \frac{\log{x}}{(1+x)^3} = -\frac12$$

Answer 2

\begin{align} &\bbox[10px,#ffd]{\int_{0}^{\infty}{\ln\left(x\right) \over \left(1 + x\right)^{3}}\,{\rm d}x} = \int_{0}^{\pi/2} {\ln\left(\tan^{2}\left(x\right)\right) \over \left\lbrack 1 + \tan^{2}\left(x\right)\right\rbrack^{3} } \,2\tan\left(x\right)\sec^{2}\left(x\right)\,{\rm d}x \\[5mm] = &\ 4\int_{0}^{\pi/2} \ln\left(\tan\left(x\right)\right) \tan\left(x\right)\cos^{4}\left(x\right)\,{\rm d}x = 4\int_{0}^{\pi/2} \ln\left(\tan\left(x\right)\right) \sin\left(x\right)\cos^{3}\left(x\right)\,{\rm d}x \\[5mm] &\ 4\int_{0}^{\pi/2} \ln\left(\sin\left(x\right)\right) \sin\left(x\right)\cos^{3}\left(x\right)\,{\rm d}x - 4\int_{0}^{\pi/2} \ln\left(\cos\left(x\right)\right) \sin\left(x\right)\cos^{3}\left(x\right)\,{\rm d}x \\[5mm] = &\ 4\int_{0}^{1} x\left(1 - x^{2}\right)\ln\left(x\right)\,{\rm d}x + 4\int_{1}^{0}x^{3}\ln\left(x\right)\,{\rm d}x = 4\int_{0}^{1} \left(x - 2x^{3}\right)\ln\left(x\right)\,{\rm d}x \\[5mm] = &\ 4\lim_{n \to 0}{{\rm d} \over {\rm d}n} \int_{0}^{1} \left(x^{n + 1} - 2x^{n + 3}\right)\,{\rm d}x = 4\lim_{n \to 0}{{\rm d} \over {\rm d}n} \left({1 \over n + 2} - {2 \over n + 4}\right) \\[5mm] = &\ 4\lim_{n \to 0} \left\lbrack -\,{1 \over \left(n + 2\right)^{2}} + {2 \over \left(n + 4\right)^{2}} \right\rbrack = 4 \left(-\,{1 \over 4} + {1 \over 8}\right) = -\,{1 \over 2} \end{align}

Answer 3

This approach is not using residues method but I'd like to post the general solution.

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Let $$\mathcal{I}=\int_0^\infty\frac{x^{m-1}}{(1+x)^{m+n}}\ln x\ dx\tag1$$ Consider beta function $$ \text{B}(m,n)=\int_0^\infty\frac{x^{m-1}}{(1+x)^{m+n}}\ dx.\tag2 $$ Differentiating $(2)$ with respect to $m$ yields \begin{align} \frac{\partial}{\partial m}\text{B}(m,n)&=\int_0^\infty\frac{\partial}{\partial m}\left(\frac{x^{m-1}}{(1+x)^{m+n}}\right)\ dx\\ (\psi(m)-\psi(m+n))\text{B}(m,n)&=\int_0^\infty\frac{x^{m-1}}{(1+x)^{m+n}}\ln\left(\frac{x}{1+x}\right)\ dx\tag3\\ &=\mathcal{I}-\color{red}{\int_0^\infty\frac{x^{m-1}}{(1+x)^{m+n}}\ln (1+x)\ dx},\tag4 \end{align} where $\psi(\cdot)$ is the digamma function.

Setting $\color{red}{\displaystyle\ x=\frac1t\;\Rightarrow\;dx=-\frac{dt}{t^2}}$ to the second integral in $(4)$ yields $$ \int_0^\infty\frac{t^{n-1}}{(1+t)^{n+m}}\ln \left(\frac{1+t}{t}\right)\ dt=(\psi(m+n)-\psi(n))\text{B}(m,n).\tag5 $$ Plugging in $(5)$ to $(4)$ yields $$ \color{blue}{\int_0^\infty\frac{x^{m-1}}{(1+x)^{m+n}}\ln x\ dx=(\psi(m)-\psi(n))\text{B}(m,n)}.\tag6 $$

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Thus, using $(6)$ and setting $m=1\; ;\; n=2$, we obtain $$ \large\int_0^\infty\frac{\ln x}{(1+x)^{3}}\ dx=(\psi(1)-\psi(2))\text{B}(1,2)=\color{blue}{-\frac12}, $$ where $\psi(1)= -\gamma$, $\ \psi(2)= 1-\gamma$, and $\displaystyle\ \text{B}(1,2)=\frac{\Gamma(1)\Gamma(2)}{\Gamma(3)}=\frac12$.