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I found the method of partial fractions very laborious to solve this definite integral : $$\int_0^\infty \frac{\sqrt[3]{x}}{1 + x^2}\,dx$$

Is there a simpler way to do this ?

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6  
You could use a keyhole contour, (see en.wikipedia.org/wiki/…) and choose $[0,\infty)$ as the branch cut for Log. However this may or may not be "simpler" depending on your definition, and how comfortable you are with complex analytic techniques. –  Eric Naslund Apr 21 '11 at 19:13
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Thanks. That was helpful. I haven't studied contour integration yet, but now I know I should. –  Balaji Rao Apr 21 '11 at 19:26
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Sorry to be picky, but you don't solve integrals, you evaluate them. –  Stefan Smith Jun 9 '12 at 2:18
    
Related to: math.stackexchange.com/questions/373164/… –  Eric Naslund Apr 26 '13 at 5:57

6 Answers 6

up vote 27 down vote accepted

Perhaps this is simpler.

Make the substitution $\displaystyle x^{2/3} = t$. Giving us

$\displaystyle \frac{2 x^{1/3}}{3 x^{2/3}} dx = dt$, i.e $\displaystyle x^{1/3} dx = \frac{3}{2} t dt$

This gives us that the integral is

$$I = \frac{3}{2} \int_{0}^{\infty} \frac{t}{1 + t^3} \ \text{d}t$$

Now make the substitution $t = \frac{1}{z}$ to get

$$I = \frac{3}{2} \int_{0}^{\infty} \frac{1}{1 + t^3} \ \text{d}t$$

Add them up, cancel the $\displaystyle 1+t$, write the denominator ($\displaystyle t^2 - t + 1$) as $\displaystyle (t+a)^2 + b^2$ and get the answer.

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Nice solution, +1. Though what you did is much cleaner, it is interesting to note that with some work you can actually write down the exact anti derivative for $\frac{1}{1+x^3}.$ (Of course it uses partial fractions, which the OP didn't want, and has all a few logs and artans and is a bit messy) –  Eric Naslund Apr 21 '11 at 19:57
    
Thank you. That was an excellent solution! –  Balaji Rao Apr 22 '11 at 4:48
    
Very nice solution, +1. –  Américo Tavares May 6 '11 at 10:05
    
Very clever, congratulations! –  Giuseppe Negro Apr 8 '12 at 19:20

Here is a different way I am quite fond of:

Call our integral I, that is set $$I=\int_0^\infty \frac{\sqrt[3]{x}}{1+x^2}\,dx$$ Let $u=1+x^{2}$ so that $du=2x \, dx$. Since $$\sqrt[3]{x} \, dx=\frac{1}{2}\frac{2x \, dx}{\sqrt[3]{x^{2}}}=\frac{1}{2}\frac{u}{\left({u-1}\right)^{\frac{1}{3}}}\,du$$ we have that$$I=\frac{1}{2}\int_{1}^{\infty}\frac{1}{u\left(u-1\right)^{\frac{1}{3}}}\,du.$$ Let $u=\frac{1}{v}$ so that this becomes $$\frac{1}{2}\int_{0}^{1}\frac{1}{\frac{1}{v}\left(\frac{1}{v}-1\right)^{\frac{1}{3}}}\frac{1}{v^{2}}\,dv=\frac{1}{2}\int_{0}^{1}v^{-\frac{2}{3}}\left(1-v\right)^{-\frac{1}{3}}\,dv=\frac{1}{2}\text{B}\left(\frac{1}{3},\frac{2}{3}\right)$$ where $\text{B}(x,y)$ is the beta function. Since $$\text{B}(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$$ we have that $$I=\frac{1}{2}\frac{\Gamma(\frac{1}{3})\Gamma(\frac{2}{3})}{\Gamma(1)}.$$ Since $\Gamma(s)\Gamma(1-s)=\frac{\pi}{\sin\pi s}$ it follows that $$I=\frac{\pi}{2\sin\pi/3}=\frac{\pi}{\sqrt{3}}.$$ Hope that helps,

Also it is worth mentioning that numerically, Wolfram Alpha agrees with this answer.

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Nice! Two quibbles: add $\mathrm{d}u$ in your first change of variables; and the established notation for the Beta function is B (standing for a capital beta). –  Did Apr 21 '11 at 20:08
    
Maybe you want to include after stating the Beta function, that it contains the gamma function which is in the fractions so people can recognize the properties that the gamma function holds. –  night owl May 6 '11 at 9:54
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@owl: The "Since $B(x,y)=\dots$" looks pretty explicit to me. –  J. M. May 6 '11 at 10:25
    
I have used both this method and residues for this kind of problem. I like them both! (+1) –  robjohn Aug 11 '12 at 13:41
    
It looks like we use a similar approach. +1. –  Tunk-Fey May 31 at 14:16

Using the same technique as in my previous answer we can generalize, and find the Mellin Transform:

Consider $$I(\alpha,\beta)=\int_{0}^{\infty}\frac{u^{\alpha-1}}{1+u^{\beta}}du=\mathcal{M}\left(\frac{1}{1+u^{\beta}}\right)(\alpha)$$ Let $x=1+u^{\beta}$ so that $u=(x-1)^{\frac{1}{\beta}}$. Then we have $$I(\alpha,\beta)=\frac{1}{\beta}\int_{1}^{\infty}\frac{(x-1)^{\frac{\alpha-1}{\beta}}}{x}(x-1)^{\frac{1}{\beta}-1}dx.$$ Setting $x=\frac{1}{v}$ we obtain $$I(\alpha,\beta)=\frac{1}{\beta}\int_{0}^{1}v^{-\frac{\alpha}{\beta}}(1-v)^{\frac{\alpha}{\beta}-1}dv=\frac{1}{\beta}\text{B}\left(-\frac{\alpha}{\beta}+1,\ \frac{\alpha}{\beta}\right).$$

Using the properties of the Beta and Gamma functions, this equals $$\frac{1}{\beta}\frac{\Gamma\left(1-\frac{\alpha}{\beta}\right)\Gamma\left(\frac{\alpha}{\beta}\right)}{\Gamma(1)}=\frac{\pi}{\beta\sin\left(\frac{\pi\alpha}{\beta}\right)}.$$

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By using techniques of complex analysis ($\text{Residue Theory}$) one can actually show that $$\int\limits_{0}^{\infty} \frac{x^{a-1}}{1+x^{b}} \ \text{dx} = \frac{\pi}{b \sin(\pi{a}/b)}, \qquad 0 < a <b$$

You can obtain the value of your $\text{Integral}$ by putting $a=\frac{4}{3}$ and $b=2$.

Set $$I = \int\limits_{0}^{\infty} \frac{x^{a-1}}{1+x^{b}} \ \text{dx}$$ and integrate $$f(z) = \frac{z^{a-1}}{1+z^{b}} = \frac{|z|^{a-1} \cdot e^{i(a-1)\text{arg}(z)}}{1+|z|^{b}e^{ib\text{arg}(z)}}$$

Simple pole at $z_{1} = e^{\pi{i}/b}$ and hence $$\text{Res} \Biggl[\frac{z^{a-1}}{1+z^{b}}, e^{\pi{i}/b}\Biggr] = \frac{z^{a-1}}{bz^{b-1}}\Biggl|_{z =e^{\pi i / b}} = -\frac{1}{b}e^{\pi i a/b}$$

Integrate along $\gamma_{1}$, and let $R \to \infty$ and let $ \epsilon \to 0^{+}$. This gives, \begin{align*} \int\limits_{\gamma_{1}} f(z) \ dz & = \int\limits_{\gamma_{1}} \frac{|z|^{a-1} \cdot e^{i(a-1)\text{arg}(z)}}{1+|z|^{b}e^{ib\text{arg}(z)}} \ dz \\ &= \int\limits_{\epsilon}^{R} \frac{x^{a-1}}{1+x^{b}} \to \int\limits_{0}^{\infty} \frac{x^{a-1}}{1+x^{b}} \ dx =I \end{align*}

Integrate along $\gamma_{2}$, and let $R \to \infty$. This gives $0 < a < b$ and $$\Biggl|\int\limits_{\gamma_{2}} f(z) dz \Biggr| \leq \frac{R^{a-1}}{R^{b}-1} \cdot \frac{2\pi R}{b} \sim \frac{2 \pi}{b R^{b-a}} \to 0$$

Integrate along $\gamma_{3}$ and let $R \to \infty$ and $\epsilon \to 0^{+}$. This gives \begin{align*} \int\limits_{\gamma_{3}} f(z) \ dz &= \int\limits_{\gamma_{3}} \frac{|z|^{a-1} \cdot e^{i(a-1)\text{arg}(z)}}{1+|z|^{b}e^{ib\text{arg}(z)}} = \Biggl[\begin{array}{c} z=x e^{2\pi i/b} \\ dz=e^{2\pi i/b} \ dx \end{array}\Biggr] \\ &= \int\limits_{R}^{\epsilon} \frac{x^{a-1}e^{2\pi i(a-1)/b}}{1+x^{b}} \cdot e^{2\pi i b} \ dx \to \int\limits_{\infty}^{0} \frac{x^{a-1}e^{2\pi i(a-1)/b}}{1+x^{b}} \cdot e^{2\pi i b} \ dx \\ &= -e^{2\pi ia/b}I \end{align*}

Integrate along $\gamma_{4}$ and let $\epsilon \to 0^{+}$. This gives $0 < a <b$, $$\Biggl|\int\limits_{\gamma_{4}} f(z) \ dz \Biggr| \leq \frac{\epsilon^{a-1}}{1-\epsilon^{b}} \cdot \frac{2\pi\epsilon}{b} \sim \frac{2\pi\epsilon}{b} \to 0$$

Using the $\text{Residue Theorem}$ and letting $R \to \infty$ and $\epsilon \to 0^{+}$, we obtain that $$ I + 0 - e^{2\pi a/b}I + 0 = 2\pi i \cdot \Bigl(-\frac{1}{b} e^{\pi ia/b}\Bigr)$$ This yields, $$(e^{-\pi i a/b} - e^{\pi i a./b})I= -\frac{2\pi i}{b}$$ and hence solving for $I$, we have $$I= \frac{2\pi i}{b \cdot (e^{\pi ia/b} - e^{-\pi i a/b})}=\frac{\pi}{b \sin(\pi a/b)}$$

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Heh, I actually started out on this problem in trying to prove the reflection formulas for the Gauss Pi function and the Legendre Gamma function: $\Pi(z)\Pi(-z) = \frac{\pi z}{\sin \pi z}$ $\Gamma(1-z)\Gamma(z) = \frac{\pi}{\sin \pi z}$ So using the Beta and Gamma functions would be circular for what I'm trying to do. But interesting that you would come to what amounts to the reverse. I had thought there'd be a method using the residue theorem but I wasn't sure what contour to use. Chandrasekhar, I'm still not clear on your choice of contours...what are $\gamma_1$ and $\gamma_2$? –  Eric Feb 29 '12 at 20:39
    
Standard reverse keyhole contour that bypass the origin and wrap up the positive part of the real line twice. –  Bombyx mori Sep 1 '12 at 7:58
    
I'm not sure, @user32240, that user9413 meant that contour, as there doesn't seem to be any reason to to bypass the origin...It's a rather unusual and odd omission not to specify clearly what the contour is in such an answer. –  DonAntonio Sep 2 '12 at 2:22
    
Well, after seeing the integrals $\,\int_\epsilon^R\,$ and etc. I think you're right...why was the origin bypassed?? I don't understand –  DonAntonio Sep 2 '12 at 2:28

Let's generalize the problem. We will evaluate $$ \int_0^\infty\dfrac{x^{\large a-1}}{1+x^b}\ dx. $$ Let $$y=\dfrac{1}{1+x^b}\quad\Rightarrow\quad x=\left(\dfrac{1-y}{y}\right)^{\large\frac1b}\quad\Rightarrow\quad dx=-\left(\dfrac{1-y}{y}\right)^{\large\frac1b-1}\ \dfrac{dy}{by^2}\ ,$$ then \begin{align} \int_0^\infty\dfrac{x^{\large a-1}}{1+x^b}\ dx&=\int_0^1 y\left(\dfrac{1-y}{y}\right)^{\large\frac{a-1}b}\left(\dfrac{1-y}{y}\right)^{\large\frac1b-1}\ \dfrac{dy}{by^2}\\&=\frac1b\int_0^1y^{\large1-\frac{a}{b}-1}(1-y)^{\large\frac{a}{b}-1}\ dy, \end{align} where the last integral in RHS is Beta function. $$ \text{B}(x,y)=\int_0^1t^{\ \large x-1}\ (1-t)^{\ \large y-1}\ dt=\frac{\Gamma(x)\cdot\Gamma(y)}{\Gamma(x+y)}. $$ Hence \begin{align} \int_0^\infty\dfrac{x^{\large a-1}}{1+x^b}\ dx&=\frac1b\int_0^1y^{\large1-\frac{a}{b}-1}(1-y)^{\large\frac{a}{b}-1}\ dy\\&=\frac1b\cdot\Gamma\left(1-\frac{a}{b}\right)\cdot\Gamma\left(\frac{a}{b}\right)\\&=\large{\color{blue}{\frac{\pi}{b\sin\left(\frac{a\pi}{b}\right)}}}. \end{align} The last part uses Euler's reflection formula for Gamma function provided $0<a<b$. Thus $$ \large\int_0^\infty\dfrac{\sqrt[3]{x}}{1+x^2}\ dx=\int_0^\infty\dfrac{x^{\large\frac43-1}}{1+x^2}\ dx=\frac{\pi}{2\sin\left(\frac{2\pi}{3}\right)}=\color{blue}{\frac\pi3\sqrt{3}}. $$

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Very nice and simple!!! I used another way to solve the generalization. –  xpaul Jul 24 at 14:15

We can use the following results $$\sum_{n=-\infty}^\infty(-1)^n\frac{1}{bn+a}=\frac{\pi}{b\sin\frac{a\pi}{b}}, \frac{1}{1-x}=\sum_{n=0}^\infty x^n $$ to evaluate the generalization. In fact \begin{eqnarray} \int_0^\infty\dfrac{x^{a-1}}{1+x^b}\ dx&=&\int_0^1\frac{x^{a-1}}{1+x^b}dx+\int_0^1\frac{x^{-a-1}}{1+x^b}dx\\ &=&\sum_{n=0}^\infty(-1)^n\int_0^1(x^{bn+a-1}+x^{bn-a-1})dx\\ &=&\sum_{n=0}^\infty(-1)^n(\frac{1}{bn+a}+\frac{1}{bn-a})\\ &=&\sum_{n=-\infty}^\infty(-1)^n\frac{1}{bn+a}\\ &=&\frac{\pi}{b\sin\frac{a\pi}{b}}. \end{eqnarray}

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